Strict quantization of coadjoint orbits

TL;DR

This paper constructs strict invariant deformation quantizations of semisimple coadjoint orbits, providing explicit formulas and isomorphisms between different real forms, advancing the understanding of quantization in Lie theory.

Contribution

It introduces a family of strict invariant products on coadjoint orbits, explicitly computes the canonical element of the Shapovalov pairing, and generalizes Wick rotation to relate different real orbit quantizations.

Findings

Constructed strict G-invariant products on holomorphic functions

Derived formal deformation quantizations of coadjoint orbits

Established isomorphisms between quantizations of different real orbits

Abstract

We obtain a family of strict $\hat G$-invariant products on the space of holomorphic functions on a semisimple coadjoint orbit of a complex connected semisimple Lie group $\hat G$. By restriction, we also obtain strict $G$-invariant products $*_\hbar$ on a space $A(O)$ of certain analytic functions on a semisimple coadjoint orbit $O$ of a real connected semisimple Lie group $G$. The space $A(O)$ endowed with one of the products $*_\hbar$ is a Fr\'echet algebra, and the formal expansion of the products around $\hbar = 0$ determines a formal deformation quantization of $O$, which is of Wick type if $G$ is compact. We study a generalization of a Wick rotation, which provides isomorphisms between the quantizations obtained for different real orbits with the same complexification. Our construction relies on an explicit computation of the canonical element of the Shapovalov pairing between generalized Verma modules, and complex analytic results on the extension of holomorphic functions.

Equations

$$F_{\lambda }=\sum _{w\in \tilde{W}}{p}_{\lambda }^w({\alpha }_w{)}^{-1}{\tilde{\pi }}_{\lambda }^{+}(X_w)\otimes {\tilde{\pi }}_{\lambda }^{-}(Y_w).$$

$${\mathcal{O}}_{\lambda }=\{\xi \in {\mathfrak{g}}^{\ast }\mid \xi ={\mathrm{Ad}}_g^{\ast }\lambda \text{ for some g∈G}\}.$$

$$\omega_{\mathrm{KKS}}(X_{\mathcal{O}_{\lambda}},Y_{\mathcal{O}_{\lambda}})\big{|}_{\xi}=\xi([X,Y])$$

$${\pi }^{\ast }:{\mathcal{C}}^{\infty }(G/G_{\lambda })\to {\mathcal{C}}^{\infty }(G{)}^{G_{\lambda }},f\mapsto {\pi }^{\ast }f:=f\circ \pi$$

$$X^{\mathrm{left}}\big{|}_{g}\coloneqq\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t}\big{|}_{t=0}g\exp(tX)\in\Gamma^{\infty}(\mathrm{T}G)\,.$$

$${\mathrm{Ad}}_g:(\mathcal{U}\mathfrak{g}{)}^{\otimes k}$$

$${\pi }^{\ast }:{\mathcal{C}}^{\infty }(G/H{)}^k$$

$$(\cdot {)}^{\mathrm{left},(1,0)}:(\mathcal{U}\mathfrak{g}{)}^{\otimes k}$$

$$\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!u_{1}\mathbin{\otimes}\dots\mathbin{\otimes}u_{k}\mapsto\mathopen{}\mathclose{{}\left((f_{1},\dots,f_{k})\mapsto u_{1}^{\mathrm{left},(1,0)}f_{1}\cdot\ldots\cdot u_{k}^{\mathrm{left},(1,0)}f_{k}}\right)\,.$$

$$U_{\mathrm{inv}}$$

$$=\{\vec{u}\in (\mathcal{U}\mathfrak{g}{)}^{\otimes k}\mid {\mathrm{Ad}}_h\vec{u}-\vec{u}\in I\text{ for all h∈H}\}.$$

$${\vec{v}}^{\mathrm{left},(1,0)}\vec{f}=0\text{ and }{\vec{u}}^{\mathrm{left},(1,0)}\vec{f}\in {\mathcal{C}}^{\infty }(G{)}^H.$$

$$(Y^{\mathrm{left}}f)(g)=\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t}\Big{|}_{t=0}f(g\exp(tY))=\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t}\Big{|}_{t=0}f(g)=0\,.$$

$$(X^{\mathrm{left}}f)(gh)=\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t}\Big{|}_{t=0}f(gh\exp(tX))=\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t}\Big{|}_{t=0}f(g\exp(t\operatorname{\mathrm{Ad}}_{h}X))=((\operatorname{\mathrm{Ad}}_{h}X)^{\mathrm{left}}f)(g)$$

$$\begin{gathered} ({\vec{u}}^{\mathrm{left},(1,0)}\vec{f})(gh)=(({\mathrm{Ad}}_h\vec{u}{)}^{\mathrm{left},(1,0)}\vec{f})(g)= \\ =({\vec{u}}^{\mathrm{left},(1,0)}\vec{f})(g)+(({\mathrm{Ad}}_h\vec{u}-\vec{u}{)}^{\mathrm{left},(1,0)}\vec{f})(g)=({\vec{u}}^{\mathrm{left},(1,0)}\vec{f})(g). \end{gathered}$$

$$\tilde{\Psi }:U_{\mathrm{inv}}\to \mathrm{Map}({\mathcal{C}}^{\infty }(G/H{)}^k,{\mathcal{C}}^{\infty }(G/H)),\tilde{\Psi }(\vec{u})\vec{f}={\pi }_{\ast }({\vec{u}}^{\mathrm{left},(1,0)}({\pi }^{\ast }\vec{f})).$$

$$\Psi :((\mathcal{U}\mathfrak{g}/\mathcal{U}\mathfrak{g}\cdot \mathfrak{h}{)}^{\otimes k}{)}^H\to {\text{k-DiffOp}}_{\mathcal{H}}^G(G/H)$$

$$\Psi ([\vec{u}])\vec{f}({\mathrm{Ad}}_g^{\ast }\lambda )=(S({\mathrm{Ad}}_g{u}_1){)}_{{\mathcal{O}}_{\lambda }}^{(1,0)}{f}_1({\mathrm{Ad}}_g^{\ast }\lambda )\cdot \dots \cdot (S({\mathrm{Ad}}_g{u}_k){)}_{{\mathcal{O}}_{\lambda }}^{(1,0)}{f}_k({\mathrm{Ad}}_g^{\ast }\lambda ).$$

$$u^{\mathrm{left}}f(g)$$

$$\displaystyle=\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t_{1}}\Big{|}_{t_{1}=0}\dots\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t_{j}}\Big{|}_{t_{j}=0}f(g\exp(t_{1}X_{1})\dots\exp(t_{j}X_{j}))$$

$$\displaystyle=\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t_{1}}\Big{|}_{t_{1}=0}\dots\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t_{j}}\Big{|}_{t_{j}=0}f(\exp(t_{1}\operatorname{\mathrm{Ad}}_{g}X_{1})\dots\exp(t_{j}\operatorname{\mathrm{Ad}}_{g}X_{j})g)$$

$$=(-{\mathrm{Ad}}_g{X}_j{)}^{\mathrm{right}}\dots (-{\mathrm{Ad}}_g{X}_1{)}^{\mathrm{right}}f(g)$$

$$=(S({\mathrm{Ad}}_{g}u){)}^{\mathrm{right}}f(g)$$

$$X^{\mathrm{right}}(\pi^{*}f)(g)=\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t}\Big{|}_{t=0}\pi^{*}f(\exp(-tX)g)=\\ =\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t}\Big{|}_{t=0}f(\operatorname{\mathrm{Ad}}^{*}_{\exp(-tX)}\operatorname{\mathrm{Ad}}^{*}_{g}\lambda)=X_{\mathcal{O}_{\lambda}}f(\operatorname{\mathrm{Ad}}_{g}^{*}\lambda)=\pi^{*}(X_{\mathcal{O}_{\lambda}}f)(g)$$

$$\Psi ([\vec{u}])\vec{f}({\mathrm{Ad}}_g^{\ast }\lambda )$$

$$=u_1^{\mathrm{left},(1,0)}({\pi }^{\ast }{f}_1)(g)\cdot \dots \cdot u_k^{\mathrm{left},(1,0)}({\pi }^{\ast }{f}_k)(g)$$

$$=(S({\mathrm{Ad}}_g{u}_1){)}^{\mathrm{right},(1,0)}({\pi }^{\ast }{f}_1)(g)\cdot \dots \cdot (S({\mathrm{Ad}}_g{u}_k){)}^{\mathrm{right},(1,0)}({\pi }^{\ast }{f}_k)(g)$$

$$=(S({\mathrm{Ad}}_g{u}_1){)}_{{\mathcal{O}}_{\lambda }}^{(1,0)}{f}_1({\mathrm{Ad}}_g^{\ast }\lambda )\cdot \dots \cdot (S({\mathrm{Ad}}_g{u}_k){)}_{{\mathcal{O}}_{\lambda }}^{(1,0)}{f}_k({\mathrm{Ad}}_g^{\ast }\lambda ).$$

$$M_{\lambda }:=\mathcal{U}\mathfrak{g}{\otimes }_{\mathcal{U}({\mathfrak{b}}^{+})}\text{mathbbm}{C}_{\lambda }^{+},M_{\lambda }^{-}:=\mathcal{U}\mathfrak{g}{\otimes }_{\mathcal{U}({\mathfrak{b}}^{-})}\text{mathbbm}{C}_{-\lambda }^{-}\text{and}{M}_{\lambda }^{\ast }:=\text{mathbbm}{C}_{\lambda }^{\ast }{\otimes }_{\mathcal{U}({\mathfrak{b}}^{-})}\mathcal{U}\mathfrak{g}.$$

$$\{Y^I{H}^J{X}^K\mid I,K\in \text{mathbbm}{N}_0^k,J\in \text{mathbbm}{N}_0^r\}\text{and}\{X^K{H}^J{Y}^I\mid I,K\in \text{mathbbm}{N}_0^k,J\in \text{mathbbm}{N}_0^r\}$$

$${\pi }_{\lambda }^{-}$$

$${\pi }_{\lambda }^{+}$$

$${\pi }_{\lambda }^{\ast }$$

$${▹}_{\lambda }^{-}$$

$${▹}_{\lambda }^{+}$$

$${◃}_{\lambda }^{\ast }$$

$$⟨\cdot ,\cdot {⟩}_{\lambda }:\mathcal{U}({\mathfrak{n}}^{+})\times \mathcal{U}({\mathfrak{n}}^{-})\to \text{mathbbm}C,(u,v)\mapsto ⟨u,v{⟩}_{\lambda }:=⟨1\otimes S(u),v\otimes 1{⟩}_{\lambda }^{\prime }.$$

$$⟨u,v{⟩}_{\lambda }={\pi }_{\lambda }(S(u)v).$$

Full text

**Mathematics Subject Classification (2020): 53D55, 17B08, 32Q28, 22E46.

Key words: Formal deformation quantization, strict quantization, coadjoint orbits, Verma modules, Shapovalov pairing, Stein manifolds.**

1. Introduction

2. 1 Preliminaries

3. 1.1 Coadjoint orbits

4. 1.2 Invariant holomorphic $k$-differential operators on homogeneous spaces

5. 2 Quantizing complex coadjoint orbits

6. 2.1 Verma modules and the Shapovalov pairing

7. 2.2 Generalization to non-regular orbits

8. 2.3 The induced formal and strict products

9. 2.4 Examples

10. 3 Continuity

11. 3.1 Continuity in the reduction-topology

12. 3.2 Stein manifolds and extension of holomorphic functions

13. 3.3 Characterizing the reduction-topology

14. 4 Quantizing real coadjoint orbits

15. 4.1 Complexification

16. 4.2 Polynomials and analytic functions

17. 4.3 Formal and strict star products on real coadjoint orbits

18. 4.4 Examples: complex projective spaces and hyperbolic discs

19. 4.5 Positive linear functionals

20. 4.6 A generalized Wick rotation

21. A Proofs, $G$-finite functions, and complex structures

22. A.1 Proofs of Subsection 1.2 and Subsection 1.2

23. A.2 $G$-finite functions

24. A.3 Complex structures on real coadjoint orbits

Introduction

The quantization problem in physics asks how to associate a quantum system to a classical mechanical system, such that the classical system can be recovered from the quantum system in a classical limit. Since both systems can be studied by their observable algebras, a first step is to quantize the classical observable algebra. This algebra is usually the Poisson algebra $\mathscr{C}^{\infty}(M)$ of smooth functions on a Poisson manifold $M$. The observable algebra of a quantum mechanical system is some non-commutative ∗-algebra $\mathcal{A}$, which in many cases is obtained from a C∗-algebra. In a second step, the states of the quantum mechanical system can be obtained as normalized positive linear functionals on $\mathcal{A}$. To define their superposition, one has to represent $\mathcal{A}$ on a (pre) Hilbert space, so that the superposition of two vector states can be defined as the vector state corresponding to the sum of the two vectors.

Formal deformation quantization, as introduced in [2], has proven to be a fruitful theory for studying some aspects of the quantization problem. One views Planck’s constant $\hbar$ as a formal parameter $\hbar$ and tries to find so-called formal star products $\star$ on $\mathcal{A}=\mathscr{C}^{\infty}(M)[\!\!\;[\hbar]\!\!\;]$, which may be thought of as the infinite jet of a full solution to the quantization problem at $\hbar=0$. These star products are associative $\mathbbm{C}[\!\!\;[\hbar]\!\!\;]$-bilinear products for which $1\in\mathscr{C}^{\infty}(M)$ is a unit and which satisfy the correct classical limit. To be more precise, if $f,g\in\mathscr{C}^{\infty}(M)$ and $f\star g=\sum_{r=0}^{\infty}\hbar^{r}C_{r}(f,g)$ with operators $C_{r}\colon\mathscr{C}^{\infty}(M)\times\mathscr{C}^{\infty}(M)\to\mathscr{C}^{\infty}(M)$, then one requires $C_{0}$ to be the pointwise multiplication, $C_{0}(f,g)=fg$, and the quantization to be in the direction of the Poisson bracket, $C_{1}(f,g)-C_{1}(g,f)=\mathrm{i}\{f,g\}$. Usually one also requires the $C_{r}$ to be bidifferential operators, so that $\star$ is local and can be restricted to open subsets of $M$. Using formal power series means on the one hand that we cannot substitute $\hbar$ with the real value of Planck’s constant as required for direct physical applications, but on the other hand that we can transfer the quantization problem to algebra by neglecting analytic aspects, such as convergence of the power series. Consequently, many powerful tools become available for its study, and existence and classification results were obtained in [5, 14, 18, 36] for symplectic manifolds, whereas in the more general case of Poisson manifolds they follow from Kontsevich’s formality theorem [28]. One can also study formal star products that are equivariant with respect to the action of a Lie group, where the classification follows for example from [15].

A complete solution of the quantization problem consists of a Hilbert space $H$ together with a quantization map that associates a quantum observable, usually a self-adjoint operator on $H$, to any classical observable. This motivates the definition of a strict quantization [30, 34, 35, 37], which is some field of “nice” ∗-algebras $\mathcal{A}_{\hbar}$ (over $\mathbbm{C}$) depending “nicely” on a parameter $\hbar$ ranging over some subset of $\mathbbm{C}$, with $\mathcal{A}_{0}$ being a completion of the classical observable algebra and the deformation being in the direction of the Poisson bracket. However, strict quantizations are much harder to understand than formal deformation quantizations. There are many examples of strict quantizations in different contexts, and therefore there are several ways to formalize the above definition, i.e. specifying the parameter set and what “nice” actually means. No general existence results are known, and a classification seems completely hopeless due to the increased complexity.

There are two prominent constructions of strict quantizations. The first is due to Rieffel [37] who, using oscillatory integrals, deforms the product on a Fréchet algebra endowed with an isometric action of $\mathbbm{R}^{d}$. If the original algebra is a C∗-algebra, then Rieffel constructs a C∗-algebraic quantization. A generalization to negatively curved Kählerian Lie groups can be found in [6]. The second construction, due to Natsume, Nest, and Peter [35], essentially glues convergent versions of the Weyl product on charts to obtain a C∗-algebraic quantization. However, both methods work only for some symplectic manifolds and fail for example for the 2-sphere with its $\mathrm{SO}(3)$-invariant symplectic structure [38]. They also make crucial use of the finite dimensionality of the classical mechanical system, so it remains unclear how to apply them to quantum field theories, despite such field theories fitting into the framework of formal deformation quantization.

Another approach to strict quantization was proposed by Beiser and Waldmann in [3, 4, 40]. They start with formal deformation quantizations, which are well-understood, and try to find subalgebras on which the formal power series converge. Such subalgebras are usually defined using additional geometric structures, and can be completed with respect to a topology in which the product is continuous. This approach was carried out explicitly for star products of exponential type on possibly infinite-dimensional vector spaces [39], for the linear Poisson structure on the dual of a Lie algebra [17], and for the hyperbolic disc $\mathbbm{D}^{n}$ using an invariant star product obtained via phase space reduction [29]. See also [41] for a survey. In this paper, we extend this approach to semisimple coadjoint orbits of connected semisimple Lie groups, which gives a much larger class of geometrically interesting examples.

Coadjoint orbits play an important role in different areas of mathematics. In the representation theory of unitary Lie groups they appear e.g. in the Kirillov orbit method [27], while in symplectic geometry they are related to momentum maps. Basic examples of coadjoint orbits are hyperbolic discs and complex projective spaces, including the 2-sphere. Any coadjoint orbit $\mathcal{O}$ of a Lie group $G$ has a canonical $G$-invariant symplectic form, and if $\mathcal{O}$ is semisimple and $G$ is compact, connected, and semisimple then there is a unique compatible $G$-invariant complex structure that makes $\mathcal{O}$ a Kähler manifold.

Constructions of star products on coadjoint orbits are due to many authors [1, 8, 9, 10, 11, 19, 25, 26]. In this paper, we focus on semisimple coadjoint orbits of connected semisimple Lie groups, and the algebraic construction of Alekseev–Lachowska [1]. The canonical element $F_{\lambda}$ of the Shapovalov pairing between certain generalized Verma modules satisfies an associativity equation generalizing that of a Drinfel’d twist. This twist induces a formal product for holomorphic functions on a complex orbit and a formal star product for smooth functions on a real orbit, and those products are compatible by restriction. It is very convenient that we can pass from one setting to the other: We will mainly work in the complex setting, which is more convenient for obtaining continuity estimates, and restrict to the real setting only in the very end.

Our first result uses methods developed by Ostapenko [32] to obtain an explicit formula for the canonical element of the Shapovalov pairing for a semisimple Lie algebra $\mathfrak{g}$:

Main Theorem I**.**

The Shapovalov pairing $\langle{}\cdot{},{}\cdot{}\rangle^{\sim}_{\lambda}\colon\mathscr{U}({\tilde{\mathfrak{n}}}^{+})\times\mathscr{U}({\tilde{\mathfrak{n}}}^{-})\to\mathbbm{C}$ is non-degenerate if $\lambda\in\tilde{\Lambda}$, and in this case its canonical element $F_{\lambda}\in\mathscr{U}({\tilde{\mathfrak{n}}}^{+})\operatorname{\hat{\otimes}}\mathscr{U}({\tilde{\mathfrak{n}}}^{-})$ is given by

[TABLE]

The notation is explained in detail in Section 2. For now, it suffices to mention that the Shapovalov pairing is a pairing between the universal enveloping algebras of two nilpotent Lie subalgebras ${\tilde{\mathfrak{n}}}^{\pm}$ of $\mathfrak{g}$, depending on a parameter $\lambda\in\mathfrak{g}^{*}$. The sum is over a set of words $\tilde{W}$ related to the root system of $\mathfrak{g}$, the $p_{\lambda}^{w}(\alpha_{w})$ are non-zero coefficients which are defined by an explicit formula, $X_{w}$ and $Y_{w}$ are elements of $\mathscr{U}\mathfrak{g}$ and $\tilde{\pi}_{\lambda}^{\pm}$ maps these elements to $\mathscr{U}({\tilde{\mathfrak{n}}}^{\pm})$. The element $F_{\hbar}$, which induces the star product, is obtained by rescaling $\lambda$, and doing so the coefficients $p_{\mathrm{i}\lambda/\hbar}^{w}(\alpha_{w})^{-1}$ will depend rationally on $\hbar$, with a countable set of poles $P$ that accumulate only at [math]. It seems as if explicit formulas for deformation quantizations received special attention by various authors, and (0.1) provides such a formula that works in great generality.

As mentioned above, the formal expansion of $F_{\hbar}$ induces formal products in complex and real settings. Furthermore, we also obtain a family of actual (non-formal) products for holomorphic polynomial functions in the complex setting and for polynomial functions in the real setting, parametrized by $\mathbbm{C}\setminus P$, since only finitely many elements of the infinite sum defining $F_{\hbar}$ are non-zero on polynomials. All these products are $G$-invariant, and under some conditions on the Cartan subalgebra used in the construction they are also Hermitian, meaning that $\overline{f*_{\hbar}g}=\overline{g}*_{\overline{\hbar}}\overline{f}$. In the real setting and for a compact semisimple connected Lie group $G$, the formal star product is of Wick type [24] with respect to the Kähler complex structure on the coadjoint orbit, meaning that it derives the first argument only in holomorphic directions and the second argument only in antiholomorphic directions.

The next major step after constructing the star product is to use the explicit formulas to prove its continuity in the complex setting with respect to the topology of locally uniform convergence. This topology is locally convex and we can extend the product to a continuous product on the completion of the holomorphic polynomials. Using methods from analytic geometry we identify this completion with the space of holomorphic functions.

Main Theorem II**.**

*For any semisimple coadjoint orbit $\hat{\mathcal{O}}$ of a connected semisimple complex Lie group $G$, there is a family of products $\mathbin{\hat{*}_{\hbar}}\colon\mathrm{Hol}(\hat{\mathcal{O}})\times\mathrm{Hol}(\hat{\mathcal{O}})\to\mathrm{Hol}(\hat{\mathcal{O}})$ for $\hbar\in\mathbbm{C}\setminus P$, where every product $\mathbin{\hat{*}_{\hbar}}$ is $G$-invariant and continuous with respect to the topology of locally uniform convergence. The dependence of $\mathbin{\hat{*}_{\hbar}}$ on $\hbar$ is holomorphic. *

This result is certainly interesting in its own right. However, as mentioned above, we can also restrict it to real coadjoint orbits $\mathcal{O}\subseteq\smash{\hat{\mathcal{O}}}$. Denote by $\mathcal{A}(\mathcal{O})$ the class of functions on $\mathcal{O}$ that extend to holomorphic functions on $\smash{\hat{\mathcal{O}}}$ (if a function extends, its extension is unique), which contains the polynomials. We define the topology of extended locally uniform convergence on $\mathcal{A}(\mathcal{O})$ by saying that a sequence of functions in $\mathcal{A}(\mathcal{O})$ converges if the corresponding sequence of extensions converges locally uniformly, so that $\mathcal{A}(\mathcal{O})$ is homeomorphic to $\smash{\mathrm{Hol}(\hat{\mathcal{O}})}$.

Main Theorem III**.**

*For any semisimple coadjoint orbit $\mathcal{O}$ of a connected semisimple real Lie group $G$, there is a family of products $\mathbin{*_{\hbar}}\colon\mathcal{A}(\mathcal{O})\times\mathcal{A}(\mathcal{O})\to\mathcal{A}(\mathcal{O})$ for $\hbar\in\mathbbm{C}\setminus P$, where every product $\mathbin{*_{\hbar}}$ is $G$-invariant and continuous with respect to the topology of extended locally uniform convergence. The dependence of $\mathbin{*_{\hbar}}$ on $\hbar$ is holomorphic. The formal expansion of $\mathbin{*_{\hbar}}$ around [math] is a formal star product deforming the $G$-invariant symplectic form of $\mathcal{O}$. *

For the hyperbolic disc the quantum algebra $(\mathcal{A}(\mathbbm{D}^{n}),*_{\hbar})$ agrees with the algebra obtained in [29] while for the 2-sphere, $(\mathcal{A}(\mathbbm{S}^{2}),*_{\hbar})$ is the algebra considered in [16].

Since we constructed a quantization of the holomorphic functions on a complex coadjoint orbit and the restriction $\smash{\mathrm{Hol}(\hat{\mathcal{O}})}\to\mathcal{A}(\mathcal{O})$ is an isomorphism, the quantizations of different real orbits with the same complexification are related:

Main Theorem IV**.**

*If $\mathcal{O}$ and $\mathcal{O}^{\prime}$ are coadjoint orbits of real semisimple connected Lie groups with the same complexification and through one common semisimple element, then the algebras $(\mathcal{A}(\mathcal{O}),*_{\hbar})$ and $(\mathcal{A}(\mathcal{O}^{\prime}),*^{\prime}_{\hbar})$ are isomorphic. *

This isomorphism generalizes the classical Wick rotation, which can be interpreted as an isomorphism between the polynomial algebras $\operatorname{\mathrm{Pol}}(\mathbbm{CP}^{n})$ and $\operatorname{\mathrm{Pol}}(\mathbbm{D}^{n})$. However, this isomorphism does not necessarily respect the star involutions with which the algebras $\mathcal{A}(\mathcal{O})$ are equipped. In other words, the algebras $\mathcal{A}(\mathcal{O})$ and $\mathcal{A}(\mathcal{O}^{\prime})$ are isomorphic as algebras, but not necessarily as ∗-algebras.

In order to apply our quantization to physics, we should represent the Fréchet algebras $(\mathcal{A}(\mathcal{O}),*_{\hbar})$ on a Hilbert space. Given a positive linear functional we can use the GNS representation to do so. For a formal star product of Wick type all point evaluation functionals are formally positive. However, formal positivity means only that the first non-vanishing order is positive and therefore, as in this case, might not survive the passage to strict products (where the contribution of higher orders can dominate the contribution of the first order). For certain coadjoint orbits we will prove that point evaluations stay positive.

One aspect that we do not discuss in this work is the relation to geometric or Berezin–Toeplitz quantization [8, 9, 10, 11]. These theories construct a quantization by studying holomorphic sections of a quantizing line bundle over the manifold $M$. This line bundle needs to satisfy some integrality condition, which for compact $M$ means that only countably many values of $\hbar$, accumulating at [math], are allowed. The algebra $\mathscr{C}^{\infty}(M)$ is, in the limit $\hbar\to 0$, approximated by finite dimensional matrix algebras. The construction of Alekseev–Lachowska coincides with another more geometric construction of star products on semisimple coadjoint orbits by Karabegov [16, 26], if $\hbar$ is not a pole. However, Karabegov’s construction still makes sense at the poles, where it coincides with (a variant of) the Berezin–Toeplitz quantization [26]. In this sense our infinite dimensional Fréchet algebras $(\mathcal{A}(\mathcal{O}),*_{\hbar})$ interpolate between the finite dimensional Berezin–Toeplitz algebras. It could be very interesting to study this in greater detail.

Contents

In Section 1 we recall some well-known facts about coadjoint orbits. This includes the realizability of coadjoint orbits as orbits of matrix Lie groups, and a characterization of invariant multidifferential operators on homogeneous spaces. In Section 2 we introduce the Shapovalov pairing of (generalized) Verma modules and derive an explicit formula for its canonical element. From this, we obtain a product for holomorphic polynomials on complex coadjoint orbits. In Section 3 we show that this product is continuous with respect to the topology of locally uniform convergence, so that we can extend it to the completion, which consists of all holomorphic functions on the orbit. Finally, we restrict our results to real coadjoint orbits in Section 4. We will determine additional properties of the star products obtained in this way (e.g. being of Wick type or of standard ordered type), study positive linear functionals, and investigate isomorphisms of the algebras obtained for different real forms of the same complex coadjoint orbit. In Appendix A we give some remaining proofs and more details on complex structures.

Notation

In the whole paper $G$ is either a real or complex Lie group, $\mathfrak{g}$ denotes the Lie algebra of $G$, and $\mathscr{U}\mathfrak{g}$ denotes the universal enveloping algebra of $\mathfrak{g}$. In Section 2 and Section 3, $G$ is always complex. In Section 4, $G$ refers to a real Lie group and $\hat{G}$ to a complexification of $G$. $K$ denotes a compact real Lie group. Coadjoint orbits through $\lambda\in\mathfrak{g}^{*}$ are denoted by $\mathcal{O}_{\lambda}$.

We write $\mathscr{C}^{\infty}(M)$ for the smooth complex-valued functions on a manifold $M$. If $M$ is a real manifold, $\mathrm{T}M$ denotes its (real) tangent bundle (so sections of $\mathrm{T}M$ are derivations of the algebra of real-valued smooth functions on $M$). The complexification of $\mathrm{T}M$ is denoted by $\mathrm{T}^{\mathbbm{C}}M$ (so sections of $\mathrm{T}^{\mathbbm{C}}M$ are derivations of $\mathscr{C}^{\infty}(M)$). If $M$ is a complex manifold, then the holomorphic tangent bundle is denoted by $\mathrm{T}^{(1,0)}M$.

1 Preliminaries

In this section we summarize some results that are needed in the rest of this article: We review the definition of coadjoint orbits and their realizability as orbits of matrix Lie groups in Subsection 1.1. In Subsection 1.2 we introduce invariant multidifferential operators on homogeneous spaces.

1.1 Coadjoint orbits

Let $G$ be a real or complex Lie group with Lie algebra $\mathfrak{g}$. We denote the adjoint action of $G$ on $\mathfrak{g}$ by $\operatorname{\mathrm{Ad}}\colon G\to\mathrm{End}(\mathfrak{g})$. For any $g\in G$, $\operatorname{\mathrm{Ad}}_{g}\coloneqq\operatorname{\mathrm{Ad}}(g)$ is the tangent map of the conjugation $G\ni x\mapsto gxg^{-1}\in G$ by $g$. Its differential $\operatorname{\mathrm{ad}}\colon\mathfrak{g}\to\mathfrak{end}(\mathfrak{g})$ is given by the Lie bracket, $\operatorname{\mathrm{ad}}_{X}(Y)=[X,Y]$. The coadjoint action $\operatorname{\mathrm{Ad}}^{*}\colon G\to\mathrm{End}(\mathfrak{g}^{*})$ of $G$ on the dual $\mathfrak{g}^{*}$ of $\mathfrak{g}$ is defined by $\operatorname{\mathrm{Ad}}^{*}_{g}\xi=\xi\circ\operatorname{\mathrm{Ad}}_{g^{-1}}$ for $\xi\in\mathfrak{g}^{*}$.

The coadjoint orbit $\mathcal{O}_{\lambda}$ of $G$ through an element $\lambda\in\mathfrak{g}^{*}$ is defined as

[TABLE]

It is well-known that $\mathcal{O}_{\lambda}\cong G/G_{\xi}$ where $\xi\in\mathcal{O}_{\lambda}$ is any point on the coadjoint orbit and $G_{\xi}=\{g\in G\mid\operatorname{\mathrm{Ad}}_{g}^{*}\xi=\xi\}$ is the stabilizer subgroup of $\xi$. If $G$ is a real (complex) Lie group, there is a unique smooth (complex) manifold structure on $G/G_{\xi}$ that makes the projection $\pi\colon G\to G/G_{\xi}$ a smooth (holomorphic) submersion, and we use it to define the structure of a smooth (complex) manifold on $\mathcal{O}_{\lambda}$. It does not depend on the choice of $\xi\in\mathcal{O}_{\lambda}$.

Fix a basis $e_{1},\dots,e_{n}$ of $\mathfrak{g}$ and let $\smash{C_{ij}^{k}}$ be the structure constants with respect to this basis, i.e. $[e_{i},e_{j}]=\sum_{k=1}^{n}C_{ij}^{k}e_{k}$. Then $\{f,g\}(\xi)=\sum_{i,j,k=1}^{n}C_{ij}^{k}\xi(e_{k})\frac{\partial f}{\partial e_{i}}\frac{\partial g}{\partial e_{j}}$ defines a linear Poisson structure on $\mathfrak{g}^{*}$, where $f,g\in\mathscr{C}^{\infty}(\mathfrak{g}^{*})$ and the $e_{i}$ are viewed as global linear coordinates on $\mathfrak{g}^{*}$. The following proposition is well-known, see e.g. [12, Example 1.1.3].

Proposition \theproposition.

*If the Lie group $G$ is connected, then the coadjoint orbits of $G$ are precisely the symplectic leaves of this linear Poisson structure. In particular, all connected Lie groups with the same Lie algebra have the same coadjoint orbits. *

Corollary \thecorollary.

*If the Lie group $G$ is semisimple and connected, then $G$ and its image under $\operatorname{\mathrm{Ad}}\colon G\to\mathrm{End}(\mathfrak{g})$ have the same coadjoint orbits. *

Proof**:**

Since $\mathfrak{g}$ is semisimple, it has trivial center and therefore $\operatorname{\mathrm{ad}}\colon\mathfrak{g}\to\mathfrak{end}(\mathfrak{g})$ is injective. Consequently, $G$ and its image in $\mathrm{End}(\mathfrak{g})$ have the same Lie algebra. Since both are connected, the result follows by applying the previous proposition. $\boxempty$

It is easy to show that not only $G$ and its image under $\operatorname{\mathrm{Ad}}$ have the same coadjoint orbits, but also $\operatorname{\mathrm{Ad}}\colon G\to\mathrm{End}(\mathfrak{g})$ intertwines the actions of $G$ and its image on the coadjoint orbits. Since the image of $G$ under $\operatorname{\mathrm{Ad}}$ is a matrix Lie group, we can therefore, when studying coadjoint orbits of connected semisimple Lie groups, assume without loss of generality that such a Lie group is a matrix Lie group. Using the argument provided in [20, Theorem 9] we can even assume that $G$ is a closed matrix Lie group.

For $X\in\mathfrak{g}$, denote the fundamental vector field of $X$ for the coadjoint action by \smash{X_{\mathcal{O}_{\lambda}}\big{|}_{\xi}}\coloneqq\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t}\big{|}_{t=0}\operatorname{\mathrm{Ad}}^{*}_{\exp(-tX)}\xi, where $\xi\in\mathcal{O}_{\lambda}$. Note that the map $\mathfrak{g}/\mathfrak{g}_{\xi}\to\mathrm{T}_{\xi}\mathcal{O}_{\lambda}$, X\mapsto X_{\mathcal{O}_{\lambda}}\big{|}_{\xi} is an isomorphism, where $\mathfrak{g}_{\xi}$ denotes the Lie algebra of $G_{\xi}$. Consequently,

[TABLE]

determines a well-defined $2$-form on $\mathcal{O}_{\lambda}$, which is called the Kirillov-Kostant-Souriau form. One can show that $\omega_{\mathrm{KKS}}$ is symplectic and $G$-invariant. By symplectic we mean that $\omega_{\mathrm{KKS}}$ is closed and that \omega_{\mathrm{KKS}}\big{|}_{\xi}\colon\mathrm{T}_{\xi}\mathcal{O}_{\lambda}\times\mathrm{T}_{\xi}\mathcal{O}_{\lambda}\to\mathbbm{k} is $\mathbbm{k}$-bilinear, antisymmetric, and non-degenerate for all $\xi\in\mathcal{O}_{\lambda}$, where $\mathbbm{k}$ is either $\mathbbm{R}$ or $\mathbbm{C}$, depending on whether $G$ is real or complex.

For a semisimple Lie algebra $\mathfrak{g}$, the Killing form $B\colon\mathfrak{g}\times\mathfrak{g}\to\mathbbm{k}$ is non-degenerate, giving an isomorphism ${}^{\flat}\colon\mathfrak{g}\to\mathfrak{g}^{*}$, $X\mapsto X^{\flat}\coloneqq B(X,{}\cdot{})$. We denote its inverse by ${}^{\sharp}\colon\mathfrak{g}^{*}\to\mathfrak{g}$. In the complex case we say that $\lambda\in\mathfrak{g}^{*}$ is semisimple if $\operatorname{\mathrm{ad}}_{\lambda^{\sharp}}\in\mathfrak{end}(\mathfrak{g})$ is diagonalisable and in the real case $\lambda\in\mathfrak{g}^{*}$ is semisimple if the complex linear extension of $\lambda$ to the complexification of $\mathfrak{g}$ is semisimple. A coadjoint orbit $\mathcal{O}_{\lambda}$ is semisimple if $\lambda$ is semisimple.

Proposition \theproposition.

*Let $G$ be a complex connected semisimple Lie group and $\lambda\in\mathfrak{g}^{*}$ be semisimple. Then $G_{\lambda}$ is connected. *

Proof**:**

The Lie algebra spanned by $\lambda^{\sharp}$ integrates to a connected commutative Lie subgroup $T^{\prime}$ of $G$, and since $\lambda^{\sharp}$ is semisimple, all elements of $T^{\prime}$ are diagonalisable in the adjoint representation. There is a smallest closed complex Lie group $T$ containing $T^{\prime}$, that can be obtained as follows: Take the closure of $T^{\prime}$ (which is a real Lie group), take the Lie algebra of this closure (which is a real Lie subalgebra of $\mathfrak{g}$), take the complex Lie algebra spanned by it, integrate this Lie algebra to a connected Lie subgroup of $G$, and possibly repeat these steps. $T$ is still connected and commutative, and all its elements are diagonalisable in the adjoint representation, so $T$ is a complex torus in $G$. Its centralizer is exactly $G_{\lambda}$, and centralizers of tori are connected. $\boxempty$

Note that the statement is also true for a real compact connected semisimple Lie group $K$, but might fail if the compactness assumption is dropped.

We denote the smooth functions on $G$ that are invariant under the action of $G_{\lambda}$ from the right by $\mathscr{C}^{\infty}(G)^{G_{\lambda}}$. That is, $f\in\mathscr{C}^{\infty}(G)^{G_{\lambda}}$ if and only if $f\in\mathscr{C}^{\infty}(G)$ and $f(gg^{\prime})=f(g)$ for all $g\in G$ and $g^{\prime}\in G_{\lambda}$. There is an algebra isomorphism

[TABLE]

and for a complex Lie group, this isomorphism restricts to an isomorphism on holomorphic functions. We denote the inverse by $\pi_{*}\colon\mathscr{C}^{\infty}(G)^{G_{\lambda}}\to\mathscr{C}^{\infty}(G/G_{\lambda})$.

Remark \theremark.

This article is written mainly from a differential geometric perspective. Note however, that any complex connected semisimple Lie group $G$ has a unique structure of an algebraic group, see Theorem 6.3 and the preceding corollary in Chapter 1 of [31]. Any holomorphic representation of $G$ is polynomial. Consequently, if $G$ is realized as a subgroup of $\mathrm{GL}_{N}(\mathbbm{C})$ it is automatically closed. The coadjoint action $G\times\mathfrak{g}^{*}\to\mathfrak{g}^{*}$ is a morphism of algebraic varieties, and coadjoint orbits of $G$ are smooth subvarieties of $\mathfrak{g}^{*}$. A coadjoint orbit of $G$ is closed in the Zariski topology if and only if it is semisimple, see [13, Theorem 5.4]. In particular, semisimple coadjoint orbits of complex connected semisimple Lie groups are affine algebraic varieties.

Note however, that this is not necessarily true for real connected semisimple Lie groups (not even if they are linear). It is still true that real connected semisimple linear Lie groups and their coadjoint orbits are connected components (with respect to the usual topology) of affine algebraic varieties.

1.2 Invariant holomorphic $k$-differential operators on homogeneous spaces

In the whole subsection $G$ is a complex Lie group, $H$ is a closed complex Lie subgroup of $G$, and $k\geq 1$ is an integer. We present some results on holomorphic $G$-invariant $k$-differential operators on the homogeneous space $G/H$, in particular we construct a bijection between the set $((\mathscr{U}\mathfrak{g}/\mathscr{U}\mathfrak{g}\cdot\mathfrak{h})^{\mathbin{\otimes}k})^{H}$ and the set of such operators. The results seem to be well-known, but proofs are hard to find in the literature.

A $k$-differential operator $D$ (see Appendix A.1 for a short review of the definition) on a manifold $M$ endowed with an action of a Lie group $G$ is said to be invariant under $G$ if $\smash{\phi_{g}^{*}}(D\smash{\vec{f}})=D((\phi_{g}^{*})^{\times k}\smash{\vec{f}})$ for all $\vec{f}\in\mathscr{C}^{\infty}(M)^{k}$ and all $g\in G$. Here $\phi_{g}\colon M\to M$ is the diffeomorphism of $M$ given by the action of a fixed element $g\in G$, and the upper star denotes the pullback. We write \smash{\text{k$-$\operatorname{\mathrm{DiffOp}}}_{\mathcal{H}}^{G}(M)} for the space of holomorphic $G$-invariant $k$-differential operators on a complex manifold $M$. A $k$-differential operator on $G$ is said to be left-invariant if it is invariant with respect to the left action $\mathrm{L}\colon G\times G\to G$, $(g,g^{\prime})\mapsto gg^{\prime}\eqqcolon\mathrm{L}_{g}(g^{\prime})$.

Let $M$ be a complex manifold with complex structure $I\colon\mathrm{T}M\to\mathrm{T}M$. For a vector field $V\in\Gamma^{\infty}(\mathrm{T}M)$ its holomorphic part is $V^{(1,0)}=\frac{1}{2}(V-\mathrm{i}IV)\in\Gamma^{\infty}(\mathrm{T}^{(1,0)}M)$. Let $\mathfrak{g}$ be the Lie algebra of $G$. For any $X\in\mathfrak{g}$ define the left-invariant vector field

[TABLE]

Its holomorphic part $X^{\mathrm{left},(1,0)}=\frac{1}{2}(X^{\mathrm{left}}-\mathrm{i}(\mathrm{i}X)^{\mathrm{left}})\in\Gamma^{\infty}(\mathrm{T}^{(1,0)}G)$ induces a holomorphic left-invariant $1$-differential operator $f\mapsto X^{\mathrm{left},(1,0)}f$ on $G$. The map $({}\cdot{})^{\mathrm{left},(1,0)}\colon\mathfrak{g}\to\Gamma^{\infty}(\mathrm{T}^{(1,0)}G)$ is a Lie algebra homomorphism, inducing an algebra homomorphism $({}\cdot{})^{\mathrm{left},(1,0)}\colon\mathscr{U}\mathfrak{g}\to\operatorname{\mathrm{DiffOp}}_{\mathcal{H}}^{G}(G)$.

In the following we extend various maps to $k$-fold products and still denote them by the same symbol,

[TABLE]

Proposition \theproposition.

*The map ({}\cdot{})^{\mathrm{left},(1,0)}\colon(\mathscr{U}\mathfrak{g})^{\mathbin{\otimes}k}\to\text{k$-$\operatorname{\mathrm{DiffOp}}}_{\mathcal{H}}^{G}(G) is an isomorphism. *

Proof**:**

See Appendix A.1. $\boxempty$

Next, we want to describe holomorphic $G$-invariant $k$-differential operators on the homogeneous space $G/H$. Let $H$ be a closed Lie subgroup of $G$ with Lie algebra $\mathfrak{h}$, and let $\mathscr{U}\mathfrak{g}\cdot\mathfrak{h}\subseteq\mathscr{U}\mathfrak{g}$ be the left ideal generated by $\mathfrak{h}$. Note that $(\mathscr{U}\mathfrak{g}/\mathscr{U}\mathfrak{g}\cdot\mathfrak{h})^{\mathbin{\otimes}k}$ is isomorphic to $(\mathscr{U}\mathfrak{g})^{\mathbin{\otimes}k}/I$ where $I=I_{1}+\dots+I_{k}$ and $I_{i}=(\mathscr{U}\mathfrak{g})^{\mathbin{\otimes}(i-1)}\mathbin{\otimes}\mathscr{U}\mathfrak{g}\cdot\mathfrak{h}\mathbin{\otimes}(\mathscr{U}\mathfrak{g})^{\mathbin{\otimes}(k-i)}$ is a left ideal in $(\mathscr{U}\mathfrak{g})^{\mathbin{\otimes}k}$. Introduce the set

[TABLE]

Here the action of $H$ on $(\mathscr{U}\mathfrak{g})^{\mathbin{\otimes}k}$ is the diagonal action defined in (1.5a).

Lemma 1.1**.**

Let $\vec{u}\in U_{\mathrm{inv}}$, $\vec{v}\in I$, and $\vec{f}\in(\mathscr{C}^{\infty}(G)^{H})^{k}$. Then

[TABLE]

Proof**:**

Let $Y\in\mathfrak{h}$ and $f\in\mathscr{C}^{\infty}(G)^{H}$. Then we compute

[TABLE]

By using that $Y^{\mathrm{left},(1,0)}=\frac{1}{2}(Y^{\mathrm{left}}-\mathrm{i}(\mathrm{i}Y)^{\mathrm{left}})$ this implies that $Y^{\mathrm{left},(1,0)}f=0$, and therefore also $\vec{v}^{\mathrm{left},(1,0)}\vec{f}=0$ for all $\vec{v}\in I$ and $\vec{f}\in(\mathscr{C}^{\infty}(G)^{H})^{k}$. If $X\in\mathfrak{g}$, then

[TABLE]

for all $f\in\mathscr{C}^{\infty}(G)^{H}$, $g\in G$, and $h\in H$. Consequently, we obtain $(X^{\mathrm{left},(1,0)}f)(gh)=((\operatorname{\mathrm{Ad}}_{h}X)^{\mathrm{left},(1,0)}f)(g)$, and extending to the universal enveloping algebra and to tensor products yields $(\vec{u}^{\mathrm{left},(1,0)}\vec{f})(gh)=((\operatorname{\mathrm{Ad}}_{h}\vec{u})^{\mathrm{left},(1,0)}\vec{f})(g)$ for all $\vec{u}\in(\mathscr{U}\mathfrak{g})^{\mathbin{\otimes}k}$ and $\vec{f}\in(\mathscr{C}^{\infty}(G)^{H})^{k}$. If $\vec{u}\in U_{\mathrm{inv}}$, then together with the first part we obtain

[TABLE]

$\boxempty$

Because of this lemma we can define

[TABLE]

Since $\pi^{*}$ and $\pi_{*}$ are algebra homomorphisms, it follows that $\tilde{\Psi}(\vec{u})$ and $\vec{u}^{\mathrm{left},(1,0)}$ satisfy essentially the same commutation relations with the operator that multiplies a component by a smooth function. Consequently $\tilde{\Psi}(\vec{u})$ is $k$-differential and of the same order than $\vec{u}^{\mathrm{left},(1,0)}$ (see the definition of $k$-differential operators given in Subsection A.1). Moreover, $\tilde{\Psi}(\vec{u})$ is $G$-invariant, because $\pi^{*}$ and $\pi_{*}$ are $G$-equivariant and $\vec{u}^{\mathrm{left},(1,0)}$ is $G$-invariant. Since $\pi\colon G\to G/H$ is a holomorphic map, it follows that $\tilde{\Psi}(\vec{u})$ is holomorphic, and $\tilde{\Psi}$ really maps into \text{k$-$\operatorname{\mathrm{DiffOp}}}^{G}_{\mathcal{H}}(G/H). The map $\tilde{\Psi}$ descends to a map

[TABLE]

because $\tilde{\Psi}(I)=0$ according to the previous lemma.

Proposition \theproposition.

*The map $\Psi$ defined in (1.8) is an isomorphism. *

Proof**:**

The proof is given in Appendix A.1. $\boxempty$

The last result of this subsection gives a description of the $k$-differential operator $\Psi([\vec{u}])$ on the coadjoint orbit without using extensions to $G$. Let $S$ be the antipode of $\mathscr{U}\mathfrak{g}$ and extend the Lie algebra homomorphism $\mathfrak{g}\ni X\mapsto X_{\mathcal{O}_{\lambda}}\in\Gamma^{\infty}(\mathrm{T}\mathcal{O}_{\lambda})$ defined just before (1.2) to an algebra homomorphism $\mathscr{U}\mathfrak{g}\to\operatorname{\mathrm{DiffOp}}(\mathcal{O}_{\lambda})$.

Proposition \theproposition.

Let $\mathcal{O}_{\lambda}\cong G/G_{\lambda}$ be a coadjoint orbit. If $\vec{u}=u_{1}\mathbin{\otimes}\dots\mathbin{\otimes}u_{k}\in U_{\mathrm{inv}}$ and $\vec{f}=(f_{1},\dots,f_{k})\in\mathscr{C}^{\infty}(\mathcal{O}_{\lambda})^{k}$, then

[TABLE]

Proof**:**

Defining the Lie algebra homomorphism $({}\cdot{})^{\mathrm{right}}\colon\mathfrak{g}\to\Gamma^{\infty}(\mathrm{T}G)$, $X\mapsto X^{\mathrm{right}}$ with X^{\mathrm{right}}\big{|}_{g}\coloneqq\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t}\big{|}_{t=0}\exp(-tX)g and extending to $\mathscr{U}\mathfrak{g}$ as before, one checks that

[TABLE]

for $u=X_{1}\dots X_{j}\in\mathscr{U}\mathfrak{g}$ and similarly $u^{\mathrm{left},(1,0)}f(g)=(S(\operatorname{\mathrm{Ad}}_{g}u))^{\mathrm{right},(1,0)}f(g)$. Furthermore, we have

[TABLE]

for all $X\in\mathfrak{g}$, implying that $X^{\mathrm{right},(1,0)}\circ\pi^{*}=\pi^{*}\circ X^{(1,0)}_{\mathcal{O}_{\lambda}}$, and therefore that $u^{\mathrm{right},(1,0)}\circ\pi^{*}=\pi^{*}\circ u^{(1,0)}_{\mathcal{O}_{\lambda}}$ for all $u\in\mathscr{U}\mathfrak{g}$. Finally,

[TABLE]

$\boxempty$

2 Quantizing complex coadjoint orbits

In this section we construct a formal associative product for holomorphic functions on a semisimple coadjoint orbit of a complex connected semisimple Lie group, and a strict associative product for polynomials. These products are induced by a twist, which is constructed using the Shapovalov pairing between generalized Verma modules. For the convenience of the reader we first consider the special case of regular semisimple orbits in Subsection 2.1, where we introduce the Shapovalov pairing between Verma modules and compute its canonical element. In Subsection 2.2 we generalize these results to non-regular semisimple orbits. In Subsection 2.3 we describe the induced formal and strict products in detail. We consider an example in Subsection 2.4.

Later, in Section 4, we will use the results of this section to obtain star products on semisimple coadjoint orbits of real connected semisimple Lie groups. From the example considered in this section, we will then obtain strict quantizations of the hyperbolic disc and the complex projective space.

2.1 Verma modules and the Shapovalov pairing

In this subsection we introduce the Shapovalov pairing between Verma modules. In case this pairing is non-degenerate, we derive an explicit formula for its canonical element, following [32]. A similar formula in the more general setting of quantum groups was obtained recently in [33]. The results allow us to quantize regular orbits.

Let $\mathfrak{g}$ be a complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$. Recall that a root is a non-zero element $\alpha\in\mathfrak{h}^{*}$ such that \mathfrak{g}^{\alpha}\coloneqq\{X\in\mathfrak{g}\mid\operatorname{\mathrm{ad}}_{H}X=\alpha(H)X\text{ for all H\in\mathfrak{h}}\} contains a non-zero element. Denote the set of roots by $\Delta$ and choose an ordering (i.e. a subset $\Delta^{+}$ of positive roots such that, setting $\Delta^{-}\coloneqq-\Delta^{+}$, we have $\Delta^{+}\cup\Delta^{-}=\Delta$, $\Delta^{+}\cap\Delta^{-}=\emptyset$, and such that if the sum of positive roots is a root, then it is positive). Denote the simple roots (i.e. elements of $\Delta^{+}$ that cannot be written as a sum of two elements of $\Delta^{+}$) by $\Sigma$. Let ${\mathfrak{n}}^{+}$ and ${\mathfrak{n}}^{-}$ be the nilpotent Lie subalgebras of $\mathfrak{g}$ spanned by the positive and negative root spaces, respectively, and define ${\mathfrak{b}}^{+}\coloneqq\mathfrak{h}\oplus{\mathfrak{n}}^{+}$ and ${\mathfrak{b}}^{-}\coloneqq\mathfrak{h}\oplus{\mathfrak{n}}^{-}$ (the direct sum is as vector spaces, the Lie algebra structure on ${\mathfrak{b}}^{\pm}\subseteq\mathfrak{g}$ is obtained by restriction from $\mathfrak{g}$).

Note that [math] is not a root. However, it is convenient to introduce the notation $\mathfrak{g}^{0}\coloneqq\mathfrak{h}$. Then $\mathfrak{g}$ is $(\Delta\cup\{0\})$-graded, in the sense that $\mathfrak{g}=\bigoplus_{\alpha\in\Delta\cup\{0\}}\mathfrak{g}^{\alpha}$ and $[\mathfrak{g}^{\alpha},\mathfrak{g}^{\beta}]\subseteq\mathfrak{g}^{\alpha+\beta}$ for any $\alpha,\beta\in\Delta\cup\{0\}$. Consequently the tensor algebra $\mathrm{T}\mathfrak{g}$ is $\mathbbm{Z}\Delta$-graded, where the so-called root lattice $\mathbbm{Z}\Delta$ is the set of linear combinations of roots. The two-sided ideal generated by elements of the form $X\mathbin{\otimes}Y-Y\mathbin{\otimes}X-[X,Y]$ with $X,Y\in\mathfrak{g}$ is homogeneous and therefore the universal enveloping algebra $\mathscr{U}\mathfrak{g}=\mathrm{T}\mathfrak{g}/\langle X\mathbin{\otimes}Y-Y\mathbin{\otimes}X-[X,Y]\rangle$ is also $\mathbbm{Z}\Delta$-graded. Denote the degree of a homogeneous element $w\in\mathscr{U}\mathfrak{g}$ by $d(w)\in\mathbbm{Z}\Delta$.

Given a linear functional $\lambda\in\mathfrak{h}^{*}$, the formula $H\mathbin{\triangleright}z=\lambda(H)z$ makes $\mathbbm{C}$ a left $\mathfrak{h}$-module, and since $\mathfrak{h}$ is commutative also a right $\mathfrak{h}$-module. We can extend this to a left or right ${\mathfrak{b}}^{\pm}$-module by noting that ${\mathfrak{b}}^{\pm}=\mathfrak{h}\oplus{\mathfrak{n}}^{\pm}$ and letting ${\mathfrak{n}}^{\pm}$ act trivially. Denote the corresponding left $\mathscr{U}({\mathfrak{b}}^{\pm})$-module by $\mathbbm{C}^{\pm}_{\lambda}$ and the right $\mathscr{U}({\mathfrak{b}}^{-})$-module by $\mathbbm{C}^{*}_{\lambda}$. Define the Verma modules

[TABLE]

Note that $M_{\lambda}$ and $M_{\lambda}^{-}$ are left $\mathscr{U}\mathfrak{g}$-modules, whereas $M_{\lambda}^{*}$ is a right $\mathscr{U}\mathfrak{g}$-module. $M_{\lambda}$ is the most general left $\mathscr{U}\mathfrak{g}$-module of highest weight $\lambda$, meaning that any other left $\mathscr{U}\mathfrak{g}$-module of highest weight $\lambda$ can be obtained as a quotient of $M_{\lambda}$. $M_{\lambda}^{-}$ is the most general left $\mathscr{U}\mathfrak{g}$-module of lowest weight $-\lambda$.

There are canonical isomorphisms $M_{\lambda}^{*}\mathbin{\otimes}_{\mathscr{U}\mathfrak{g}}M_{\lambda}\cong\mathbbm{C}^{*}_{\lambda}\mathbin{\otimes}_{\mathscr{U}({\mathfrak{b}}^{-})}\mathscr{U}\mathfrak{g}\mathbin{\otimes}_{\mathscr{U}({\mathfrak{b}}^{+})}\mathbbm{C}_{\lambda}\cong\mathbbm{C}^{*}_{\lambda}\mathbin{\otimes}_{\mathscr{U}\mathfrak{h}}\mathbbm{C}_{\lambda}\cong\mathbbm{C}$ since the left and right $\mathfrak{h}$-module structures on $\mathbbm{C}$ coincide.

Definition \thedefinition.

*The pairing $\langle{}\cdot{},{}\cdot{}\rangle^{\prime}_{\lambda}\colon M^{*}_{\lambda}\times M_{\lambda}\to\mathbbm{C}$ defined by $(x,y)\mapsto x\mathbin{\otimes}_{\mathscr{U}\mathfrak{g}}y$ is called the Shapovalov pairing between $M^{*}_{\lambda}$ and $M_{\lambda}$. *

In the following it will be convenient to have alternative descriptions of $M_{\lambda}$, $M_{\lambda}^{-}$ and $M^{*}_{\lambda}$. Let $\{X_{1},\dots,X_{k}\}$ be a basis of ${\mathfrak{n}}^{+}$, $\{Y_{1},\dots,Y_{k}\}$ be a basis of ${\mathfrak{n}}^{-}$, and $\{H_{1},\dots,H_{r}\}$ be a basis of $\mathfrak{h}$. Since $\mathfrak{g}={\mathfrak{n}}^{+}\oplus\mathfrak{h}\oplus{\mathfrak{n}}^{-}$ (as vector spaces) the Poincaré–Birkhoff–Witt theorem implies that

[TABLE]

are bases for $\mathscr{U}\mathfrak{g}$. Here we use the multiindex notation $Y^{I}\coloneqq Y_{1}^{I_{1}}\dots Y_{k}^{I_{k}}$ (and similarly for $H$ and $X$). Define maps

[TABLE]

where $\delta_{K,0}$ is $1$ if $K=(0,\dots,0)$ and is [math] otherwise. Note that $\pi_{\lambda}^{\pm}$ and $\pi_{\lambda}^{*}$ are independent of the choice of bases. Fix non-zero vectors $1\in\mathbbm{C}^{\pm}_{\lambda}$ and $1\in\mathbbm{C}^{*}_{\lambda}$ (thinking of $\mathbbm{C}$ as a vector space, this choice is not canonical).

Lemma 2.1**.**

The maps ${}\cdot{}\mathbin{\otimes}1\colon\mathscr{U}({\mathfrak{n}}^{-})\to M_{\lambda}$, $v\mapsto v\mathbin{\otimes}1$ and ${}\cdot{}\mathbin{\otimes}1\colon\mathscr{U}({\mathfrak{n}}^{+})\to M^{-}_{\lambda}$, $u\mapsto u\mathbin{\otimes}1$ define isomorphisms of left $\mathscr{U}({\mathfrak{n}}^{-})$-modules and $\mathscr{U}({\mathfrak{n}}^{+})$-modules, respectively. The map $1\mathbin{\otimes}{}\cdot{}\colon\mathscr{U}({\mathfrak{n}}^{+})\to M^{*}_{\lambda}$, $u\mapsto 1\mathbin{\otimes}u$ defines an isomorphism of right $\mathscr{U}({\mathfrak{n}}^{+})$-modules. The $\mathscr{U}\mathfrak{g}$-module structures on $\mathscr{U}({\mathfrak{n}}^{\pm})$ obtained by transferring the module structures on the Verma modules with these isomorphisms are given explicitly by

[TABLE]

*Furthermore, $S(w\mathbin{\mathbin{\triangleright}_{\lambda}^{+}}u)=S(u)\mathbin{\mathbin{\triangleleft}_{\lambda}^{*}}S(w)$, where $S$ denotes the antipode of $\mathscr{U}\mathfrak{g}$. Or, in other words, $S\colon\mathscr{U}({\mathfrak{n}}^{+})\to\mathscr{U}({\mathfrak{n}}^{+})$ is an isomorphism from the left $\mathscr{U}\mathfrak{g}$-module $(\mathscr{U}({\mathfrak{n}}^{+}),\mathbin{\triangleright}_{\lambda}^{+})$ to the right $\mathscr{U}\mathfrak{g}$-module $(\mathscr{U}({\mathfrak{n}}^{+}),\mathbin{\triangleleft}_{\lambda}^{*})$ over the map $S\colon\mathscr{U}\mathfrak{g}\to\mathscr{U}\mathfrak{g}$. *

Proof**:**

One checks easily that the maps $M_{\lambda}\to\mathscr{U}({\mathfrak{n}}^{-})$, $w\mathbin{\otimes}z1\mapsto z\cdot\pi^{-}_{\lambda}(w)$ and $M^{-}_{\lambda}\to\mathscr{U}({\mathfrak{n}}^{+})$, $w\mathbin{\otimes}z1\mapsto z\cdot\pi^{+}_{\lambda}(w)$ as well as $M^{*}_{\lambda}\to\mathscr{U}({\mathfrak{n}}^{+})$, $z1\mathbin{\otimes}w\mapsto z\cdot\pi^{*}_{\lambda}(w)$ are all well-defined and inverses of the maps in the statement of the lemma. Consequently, $w\mathbin{\mathbin{\triangleright}_{\lambda}^{-}}v=({}\cdot{}\mathbin{\otimes}1)^{-1}(wv\mathbin{\otimes}1)=\pi^{-}_{\lambda}(wv)$, and (2.4b) and (2.4c) follow similarly. Finally, $\pi_{\lambda}^{*}\circ S=S\circ\pi_{\lambda}^{+}$, so $S(w\mathbin{\mathbin{\triangleright}_{\lambda}^{+}}u)=S\circ\pi_{\lambda}^{+}(wu)=\pi_{\lambda}^{*}\circ S(wu)=\pi_{\lambda}^{*}(S(u)S(w))=S(u)\mathbin{\mathbin{\triangleleft}_{\lambda}^{*}}S(w)$. $\boxempty$

The pairing of the left $\mathscr{U}\mathfrak{g}$-modules $(\mathscr{U}({\mathfrak{n}}^{\pm}),\mathbin{\triangleright}_{\lambda}^{\pm})$ obtained from the Shapovalov pairing by composing with the isomorphisms $(\mathscr{U}({\mathfrak{n}}^{-}),\mathbin{\triangleright}_{\lambda}^{-})\xrightarrow{{}\cdot{}\mathbin{\otimes}1}M_{\lambda}$ and $(\mathscr{U}({\mathfrak{n}}^{+}),\mathbin{\triangleright}_{\lambda}^{+})\xrightarrow{S}(\mathscr{U}({\mathfrak{n}}^{+}),\mathbin{\triangleleft}_{\lambda}^{*})\xrightarrow{1\mathbin{\otimes}{}\cdot{}}M_{\lambda}^{*}$ of the previous lemma, is

[TABLE]

In order to compute $\langle u,v\rangle_{\lambda}$ for $u\in\mathscr{U}({\mathfrak{n}}^{+})$ and $v\in\mathscr{U}({\mathfrak{n}}^{-})$ one needs to write $S(u)v\in\mathscr{U}\mathfrak{g}$ in the form $\sum_{i}v^{\prime}_{i}h^{\prime}_{i}u^{\prime}_{i}$ with $u^{\prime}_{i}\in\mathscr{U}({\mathfrak{n}}^{+})$, $v^{\prime}_{i}\in\mathscr{U}({\mathfrak{n}}^{-})$ and $h^{\prime}_{i}\in\mathscr{U}\mathfrak{h}$. The pairing is then given by summing $\lambda(h^{\prime}_{i})$ for those summands that have $v^{\prime}_{i}=u^{\prime}_{i}=1$. This is made more precise in the next lemma. Define $\pi_{\lambda}\coloneqq\pi_{\lambda}^{-}\circ\pi_{\lambda}^{*}=\pi_{\lambda}^{*}\circ\pi_{\lambda}^{-}\colon\mathscr{U}\mathfrak{g}\to\mathbbm{C}$, where $\mathbbm{C}$ is identified with $\mathbbm{C}1\subseteq\mathscr{U}({\mathfrak{n}}^{\pm})$ and we have implicitly used the inclusion $\mathscr{U}({\mathfrak{n}}^{\pm})\to\mathscr{U}\mathfrak{g}$ when composing the maps.

Lemma 2.2**.**

For $u\in\mathscr{U}({\mathfrak{n}}^{+})$ and $v\in\mathscr{U}({\mathfrak{n}}^{-})$ the pairing $\langle{}\cdot{},{}\cdot{}\rangle_{\lambda}$ defined in (2.5) can be computed as

[TABLE]

It is $\mathscr{U}\mathfrak{g}$-invariant, in the sense that $\langle w\mathbin{\mathbin{\triangleright}_{\lambda}^{+}}u,v\rangle_{\lambda}=\langle u,S(w)\mathbin{\mathbin{\triangleright}_{\lambda}^{-}}v\rangle_{\lambda}$ for $u\in\mathscr{U}({\mathfrak{n}}^{+})$, $v\in\mathscr{U}({\mathfrak{n}}^{-})$ and $w\in\mathscr{U}\mathfrak{g}$. The pairing respects the degree $d$ defined in the beginning of this section, meaning that $\langle u,v\rangle_{\lambda}=0$ for homogeneous elements $u\in\mathscr{U}({\mathfrak{n}}^{+})$ and $v\in\mathscr{U}({\mathfrak{n}}^{-})$ with $d(u)\neq-d(v)$. Furthermore, if $d(u)=-d(v)$, then

[TABLE]

Proof**:**

By definition $\langle u,v\rangle_{\lambda}=1\mathbin{\otimes}_{\mathscr{U}({\mathfrak{b}}^{-})}S(u)v\mathbin{\otimes}_{\mathscr{U}({\mathfrak{b}}^{+})}1$. So to prove (2.6) it suffices to check that $1\mathbin{\otimes}_{\mathscr{U}({\mathfrak{b}}^{-})}w\mathbin{\otimes}_{\mathscr{U}({\mathfrak{b}}^{+})}1=\pi_{\lambda}(w)$ for all $w\in\mathscr{U}\mathfrak{g}$, which one can easily verify on the basis $\{Y^{I}H^{J}X^{K}\mid I,K\in\mathbbm{N}_{0}^{k},J\in\mathbbm{N}_{0}^{r}\}$. The $\mathscr{U}\mathfrak{g}$-invariance follows by noting that $\langle{}\cdot{},{}\cdot{}\rangle^{\prime}_{\lambda}$ is $\mathscr{U}\mathfrak{g}$-invariant, meaning $\langle xw,y\rangle^{\prime}_{\lambda}=\langle x,wy\rangle^{\prime}_{\lambda}$ for $x\in M_{\lambda}^{*}$ and $y\in M_{\lambda}$, and using the isomorphisms of the previous lemma. For homogeneous $u\in\mathscr{U}({\mathfrak{n}}^{+})$ and $v\in\mathscr{U}({\mathfrak{n}}^{-})$ with $d(u)\neq-d(v)$ it follows that $S(u)v$ is also homogeneous of degree $d(u)+d(v)\neq 0$ and therefore $\pi_{\lambda}(S(u)v)=0$. Finally, if $d(u)=-d(v)$, then $d(S(u)v)=0$ and $\langle u,v\rangle_{\lambda}1_{\mathscr{U}({\mathfrak{n}}^{-})}=\pi_{\lambda}(S(u)v)1_{\mathscr{U}({\mathfrak{n}}^{-})}=\pi_{\lambda}^{-}(S(u)v)=S(u)\mathbin{\triangleright}_{\lambda}^{-}v$, implying the first equality of (2.7). The second one follows from applying $S$ on both sides of $\langle u,v\rangle_{\lambda}1_{\mathscr{U}({\mathfrak{n}}^{+})}=\pi_{\lambda}(S(u)v)1_{\mathscr{U}({\mathfrak{n}}^{+})}=\pi_{\lambda}^{*}(S(u)v)=S(\pi_{\lambda}^{+}(S(v)u))=S(S(v)\mathbin{\triangleright}_{\lambda}^{+}u)$. $\boxempty$

If the pairing $\langle{}\cdot{},{}\cdot{}\rangle_{\lambda}$ is non-degenerate, we can pick bases $\{u_{i}\}_{i\in\mathbbm{N}}$ of $\mathscr{U}({\mathfrak{n}}^{+})$ and $\{v_{j}\}_{j\in\mathbbm{N}}$ of $\mathscr{U}({\mathfrak{n}}^{-})$ consisting of homogeneous elements with respect to $d$ and satisfying $\langle u_{i},v_{j}\rangle_{\lambda}=\delta_{ij}$. Then the element $F_{\lambda}\coloneqq\sum_{i=1}^{\infty}u_{i}\mathbin{\otimes}v_{i}\in\mathscr{U}({\mathfrak{n}}^{+})\operatorname{\hat{\otimes}}\mathscr{U}({\mathfrak{n}}^{-})$ is called the canonical element of the pairing. It is independent of the choice of bases. By $\mathscr{U}({\mathfrak{n}}^{+})\operatorname{\hat{\otimes}}\mathscr{U}({\mathfrak{n}}^{-})$ we mean the completion of the tensor product with respect to the $\mathbbm{Z}\Delta$-grading $d$ defined in the beginning of this subsection, which is needed to make sense of the infinite sum. The following lemma is a standard statement when working with canonical elements.

Lemma 2.3**.**

Assume that $\langle{}\cdot{},{}\cdot{}\rangle_{\lambda}$ is non-degenerate, and let $F_{\lambda}=\sum_{i=1}^{\infty}u_{i}\mathbin{\otimes}v_{i}\in\mathscr{U}({\mathfrak{n}}^{+})\operatorname{\hat{\otimes}}\mathscr{U}({\mathfrak{n}}^{-})$ be its canonical element. Then

[TABLE]

*hold for all $u\in\mathscr{U}({\mathfrak{n}}^{+})$ and all $v\in\mathscr{U}({\mathfrak{n}}^{-})$, and $F_{\lambda}$ is uniquely determined by this property. *

Note that $\langle u,v_{i}\rangle$ and $\langle u_{i},v\rangle$ are non-zero for only finitely many indices $i$, so that the sums in (2.8) are both finite. The pairing $\langle{}\cdot{},{}\cdot{}\rangle_{\lambda}$ is non-degenerate precisely when the Verma modules are irreducible, but we will not need this below. In order to determine $F_{\lambda}$ explicitly, we need to introduce some more notation.

Denote the Killing form of $\mathfrak{g}$ by $B$. Since $\mathfrak{g}$ is semisimple, $B$ is non-degenerate on $\mathfrak{g}$. Extending linear functionals on $\mathfrak{h}$ by [math] on the root spaces $\mathfrak{g}^{\alpha}$, we may view $\mathfrak{h}^{*}$ as a subspace of $\mathfrak{g}^{*}$. Since $B$ restricts to zero on $\mathfrak{h}\times\mathfrak{g}^{\alpha}$ for any $\alpha\in\Delta$, it follows that $B$ is non-degenerate on $\mathfrak{h}$ and that the maps ${}^{\flat}\colon\mathfrak{g}\to\mathfrak{g}^{*}$ and ${}^{\sharp}\colon\mathfrak{g}^{*}\to\mathfrak{g}$ defined in Subsection 1.1 restrict to mutually inverse isomorphisms ${}^{\flat}\colon\mathfrak{h}\to\mathfrak{h}^{*}$ and ${}^{\sharp}\colon\mathfrak{h}^{*}\to\mathfrak{h}$. For $\alpha,\beta\in\mathfrak{h}^{*}$, let $(\alpha,\beta)\coloneqq B(\alpha^{\sharp},\beta^{\sharp})$.

Denote the positive roots by $\alpha_{1},\dots,\alpha_{k}$. For every positive root $\alpha_{i}\in\Delta^{+}$ choose elements $X_{i}\coloneqq X_{\alpha_{i}}\in\mathfrak{g}^{\alpha_{i}}$ and $Y_{i}\coloneqq Y_{\alpha_{i}}=X_{-\alpha_{i}}\in\mathfrak{g}^{-\alpha_{i}}$ such that $B(X_{i},Y_{i})=1$. Then we have $[X_{i},Y_{i}]=\alpha_{i}^{\sharp}$ since for all $H\in\mathfrak{h}$,

[TABLE]

and the Killing form is non-degenerate on $\mathfrak{h}$. Note that $[\alpha_{i}^{\sharp},X_{i}]=\alpha_{i}(\alpha_{i}^{\sharp})X_{i}=(\alpha_{i},\alpha_{i})X_{i}$ and similarly $[\alpha_{i}^{\sharp},Y_{i}]=-(\alpha_{i},\alpha_{i})Y_{i}$, so $X^{\prime}_{i}=2(\alpha_{i},\alpha_{i})^{-1}X_{i}$, $Y^{\prime}_{i}=Y_{i}$ and $H^{\prime}_{i}=2(\alpha_{i},\alpha_{i})^{-1}\alpha_{i}^{\sharp}$ satisfy the commutation relations $[X^{\prime}_{i},Y^{\prime}_{i}]=H^{\prime}_{i}$, $[H^{\prime}_{i},X^{\prime}_{i}]=2X^{\prime}_{i}$ and $[H^{\prime}_{i},Y^{\prime}_{i}]=-2Y^{\prime}_{i}$ of the usual generators of $\mathfrak{sl}_{2}(\mathbbm{C})$, the special linear Lie algebra in 2 dimensions.

Let $\rho=\frac{1}{2}\sum_{\alpha\in\Delta^{+}}\alpha$ be the half-sum of all positive roots. Denote non-negative integral linear combinations of positive roots by $\mathbbm{N}_{0}\Delta^{+}$. For $\lambda\in\mathfrak{h}^{*}$ fixed, and $\mu\in\mathfrak{h}^{*}$ define the number

[TABLE]

Recall that for a representation $\varrho\colon\mathfrak{g}\to V$ and $\mu\in\mathfrak{h}^{*}$ we define V^{\mu}\coloneqq\{v\in V\mid\varrho(H)v=\mu(H)v\text{ for all H\in\mathfrak{h}}\}. If $V^{\mu}\neq\{0\}$, then we call $\mu$ a weight and any $v\in V^{\mu}$ is called a weight vector of weight $\mu$. $V$ is called a weight module if $V=\smash{\bigoplus_{\mu\in\mathfrak{h}^{*}}V^{\mu}}$. A highest weight module is a weight module generated by a vector $v\in V$ satisfying $X_{\alpha}v=0$ for all $\alpha\in\Delta^{+}$. It is said to be of highest weight $\mu$ if $v\in V^{\mu}$.

Lemma 2.4** (Ostapenko, [32, Lemma 2]).**

Let $V$ be a highest weight module of highest weight $\lambda$, assume $\mu\in\mathbbm{N}_{0}\Delta^{+}$, and let $v\in V^{\lambda-\mu}$. Then

[TABLE]

Proof**:**

Choose an orthonormal basis $\{H_{1},\dots,H_{r}\}$ of $\mathfrak{h}$ with respect to the Killing form. The Casimir element

[TABLE]

acts as a scalar on $V$ because $V$ is generated by a highest weight vector and $c$ is central in $\mathscr{U}\mathfrak{g}$. Evaluating it on a highest weight vector the $Y_{\alpha}X_{\alpha}$-part vanishes and we obtain that $c$ acts as multiplication by $\sum_{\alpha\in\Delta^{+}}(\alpha,\lambda)+\sum_{i=1}^{r}\lambda(H_{i})\lambda(H_{i})=(2\rho,\lambda)+(\lambda,\lambda)$. Therefore

[TABLE]

holds for any $v\in V^{\lambda-\mu}$, and rearranging this equation proves the lemma. $\boxempty$

Let $W$ be the set of words with letters from $\{1,\dots,k\}$. For any $w=(w_{1},\dots,w_{\lvert w\rvert})\in W$, we define $w^{\mathrm{opp}}\coloneqq(w_{\lvert w\rvert},\dots,w_{1})$, $w_{i\dots j}\coloneqq(w_{i},\dots,w_{j})$, $X_{w}\coloneqq\smash{X_{w_{1}}\dots X_{w_{\lvert w\rvert}}}\in\mathscr{U}({\mathfrak{n}}^{+})$, $Y_{w}\coloneqq\smash{Y_{w_{1}}\dots Y_{w_{\lvert w\rvert}}}\in\mathscr{U}({\mathfrak{n}}^{-})$ and $\alpha_{w}\coloneqq\alpha_{w_{1}}+\dots+\alpha_{w_{\lvert w\rvert}}$. We use $w_{i\dots j}\coloneqq\emptyset$ if $j<i$, $X_{\emptyset}\coloneqq 1$, $Y_{\emptyset}\coloneqq 1$ and $\alpha_{\emptyset}\coloneqq 0$. Furthermore let

[TABLE]

We call a set $T$ of words a tree if $w=(w_{1},\dots,w_{\lvert w\rvert})\in T$ implies that $w_{1\dots i}\in T$ for all $i=0,\dots,\lvert w\rvert-1$ and $(w_{1},w_{2},\dots,w_{\lvert w\rvert-1},x)\in T$ for all $x\in\{1,\dots,k\}$. See Figure 1 for a visualization of a tree. For a tree $T$ we denote by $\max T$ the set of elements $w\in T$ such that $w\neq w^{\prime}_{1\dots i}$ for any $w^{\prime}\in T$ and any $i\in\{0,\dots,\lvert w^{\prime}\rvert-1\}$. Finally a tree is said to be $\mu$-admissible if $p_{\lambda}(\mu-\alpha_{w})\neq 0$ for all $w\in T\setminus\max T$, or equivalently if $p^{w}_{\lambda}(\mu)\neq 0$ for all $w\in T$.

Lemma 2.5** (Ostapenko, [32, Theorem 3]).**

Let $V$ be a highest weight module of highest weight $\lambda$, assume $\mu\in\mathbbm{N}_{0}\Delta^{+}$, and let $v\in V^{\lambda-\mu}$. Then

[TABLE]

*holds for every $\mu$-admissible tree $T$. *

Proof**:**

Apply the previous lemma repeatedly. $\boxempty$

Lemma 2.6**.**

Let $V$ be a lowest weight module of lowest weight $-\lambda$, assume $\mu\in\mathbbm{N}_{0}\Delta^{+}$, and let $v\in V^{-\lambda+\mu}$. Then $\sum_{\alpha\in\Delta^{+}}X_{\alpha}Y_{\alpha}v=-p_{\lambda}(\mu)v$, and

[TABLE]

*holds for every $\mu$-admissible tree $T$. *

Proof**:**

Similar to the proof of Lemma 2.4 and Lemma 2.5. $\boxempty$

Define the set

[TABLE]

Proposition \theproposition.

The Shapovalov pairing $\langle{}\cdot{},{}\cdot{}\rangle_{\lambda}\colon\mathscr{U}({\mathfrak{n}}^{+})\times\mathscr{U}({\mathfrak{n}}^{-})\to\mathbbm{C}$ is non-degenerate for $\lambda\in\Lambda$, and in this case its canonical element $F_{\lambda}\in\mathscr{U}({\mathfrak{n}}^{+})\operatorname{\hat{\otimes}}\mathscr{U}({\mathfrak{n}}^{-})$ is given by

[TABLE]

Proof**:**

We check that $F_{\lambda}$ satisfies the property given in Lemma 2.3. We decompose $v\in\mathscr{U}({\mathfrak{n}}^{-})$ as $v=\sum_{\mu\in\mathbbm{N}_{0}\Delta^{+}}v_{-\mu}$ where $v_{-\mu}$ is homogeneous of degree $-\mu$ with respect to the $\mathbbm{Z}\Delta$-grading. For $\mu\in\mathbbm{N}_{0}\Delta^{+}$ let $W_{\mu}$ be the set of words $w\in W$ satisfying $\alpha_{w}=\mu$. Then

[TABLE]

The first equality holds because $Y_{w}\langle X_{w},v\rangle_{\lambda}=Y_{w}\mathbin{\mathbin{\triangleright}_{\lambda}^{-}}(\langle X_{w},v_{-\alpha_{w}}\rangle_{\lambda}1_{\mathscr{U}({\mathfrak{n}}^{-})})=Y_{w}\mathbin{\mathbin{\triangleright}_{\lambda}^{-}}S(X_{w})\mathbin{\mathbin{\triangleright}_{\lambda}^{-}}v_{-\alpha_{w}}$ by Lemma 2.2. The second equality is true by basic manipulations. The third equality follows from Lemma 2.5 because we can rewrite the sum over all $w\in W_{\mu}$ as a sum over $\max T$ for a $\mu$-admissible tree $T$ as follows: Define

[TABLE]

which is the smallest tree containing $W_{\mu}$. Since $\lambda\in\Lambda$ this tree is $\mu$-admissible, and clearly $W_{\mu}\subseteq\max T$. Let $w\in\max T$. Then either $\alpha_{w}=\mu$, so that $w\in W_{\mu}$, or there do not exist $w^{\prime}\in W_{\mu}$ and $i\in\{0,\dots,\lvert w^{\prime}\rvert\}$ with $w=w^{\prime}_{1\dots i}$, so that $\mu-\alpha_{w}\notin\mathbbm{N}_{0}\Delta^{+}$ and therefore $X_{w^{\mathrm{opp}}}v_{-\mu}=0$.

Similarly, for $u=\sum_{\mu\in\mathbbm{N}_{0}\Delta^{+}}u_{\mu}\in\mathscr{U}({\mathfrak{n}}^{+})$ with $d(u_{\mu})=\mu$ we compute that

[TABLE]

using $X_{w}\langle u,Y_{w}\rangle_{\lambda}=X_{w}\mathbin{\triangleright}_{\lambda}^{+}(\langle u_{\alpha_{w}},Y_{w}\rangle_{\lambda}1_{\mathscr{U}({\mathfrak{n}}^{+})})=X_{w}\mathbin{\triangleright}^{+}_{\lambda}S(Y_{w})\mathbin{\triangleright}^{+}_{\lambda}u_{\alpha_{w}}$, and that the sum over $w\in W_{\mu}$ can be rewritten as a sum over maximal elements of a tree $T$ in a similar way as before.  $\boxempty$

Using the inclusion $\mathscr{U}({\mathfrak{n}}^{+})\operatorname{\hat{\otimes}}\mathscr{U}({\mathfrak{n}}^{-})\to(\mathscr{U}\mathfrak{g})^{\hat{\mathbin{\otimes}}2}$ and passing to the quotient, we can map the element $F_{\lambda}$ from (2.15) to $(\mathscr{U}\mathfrak{g}/\mathscr{U}\mathfrak{g}\cdot\mathfrak{h})^{\smash{\hat{\mathbin{\otimes}}2}}$. Note that $\mathscr{U}\mathfrak{g}\cdot\mathfrak{h}$ is a homogeneous ideal in $\mathscr{U}\mathfrak{g}$ with respect to the degree $d$, so the quotient $\mathscr{U}\mathfrak{g}/\mathscr{U}\mathfrak{g}\cdot\mathfrak{h}$ is still graded. The completed tensor product is defined with respect to this grading. The action of $\mathfrak{h}$ on $(\mathscr{U}\mathfrak{g})^{\mathbin{\otimes}2}$ given by $H\mathbin{\triangleright}(w\mathbin{\otimes}w^{\prime})=\operatorname{\mathrm{ad}}_{H}w\mathbin{\otimes}w^{\prime}+w\mathbin{\otimes}\operatorname{\mathrm{ad}}_{H}w^{\prime}$ with $H\in\mathfrak{h}$ and $w,w^{\prime}\in\mathscr{U}\mathfrak{g}$ stays well-defined on the quotient and preserves the degree, so it extends uniquely to a continuous action on the completed tensor product. Denote the coproduct of the Hopf algebra $\mathscr{U}\mathfrak{g}$ by $\Delta$. It is defined by extending the assignment $\mathfrak{g}\ni X\mapsto X\mathbin{\otimes}1+1\mathbin{\otimes}X\in\mathscr{U}\mathfrak{g}\mathbin{\otimes}\mathscr{U}\mathfrak{g}$ to an algebra homomorphism $\Delta\colon\mathscr{U}\mathfrak{g}\to\mathscr{U}\mathfrak{g}\mathbin{\otimes}\mathscr{U}\mathfrak{g}$.

Proposition \theproposition (Alekseev–Lachowska [1]).

Let $\lambda\in\Lambda$. Then $F_{\lambda}\in(\mathscr{U}\mathfrak{g}/\mathscr{U}\mathfrak{g}\cdot\mathfrak{h})^{\hat{\mathbin{\otimes}}2}$ is $\mathfrak{h}$-invariant and satisfies

[TABLE]

*in $(\mathscr{U}\mathfrak{g}/\mathscr{U}\mathfrak{g}\cdot\mathfrak{h})^{\hat{\mathbin{\otimes}}3}$. *

Proof**:**

See the proof of Subsection 2.3. $\boxempty$

Using the results of Subsection 1.2, elements of $((\mathscr{U}\mathfrak{g}/\mathscr{U}\mathfrak{g}\cdot\mathfrak{h})^{\mathbin{\otimes}2})^{H}$ determine bidifferential operators on a complex coadjoint orbit for which $\mathfrak{g}_{\lambda}=\mathfrak{h}$. Such orbits are of maximal dimension among all coadjoint orbits and called regular. Note that $H$ is automatically connected by Subsection 1.1, so $\mathfrak{h}$-invariance of $F_{\lambda}$ implies $H$-invariance, but $F_{\lambda}$ is only an element of the completed tensor product. So applying the construction from Subsection 1.2 naively gives a sum of bidifferential operators of increasing orders. To make sense of this sum, we can either introduce a formal parameter $\hbar$ in the construction in such a way that we obtain a formal power series of bidifferential operators, or we can restrict ourselves to applying these operators to some class of polynomials, for which only finitely many of the bidifferential operators appearing in the sum give a non-zero contribution.

We will now proceed as follows: In Subsection 2.2, we generalize the construction of $F_{\lambda}$ to work for arbitrary stabilizers $\mathfrak{g}_{\lambda}$ (and not just $\mathfrak{h}$). In Subsection 2.3 we will give details on how to construct bidifferential operators out of $F_{\lambda}$, both in the formal and polynomial settings mentioned above.

2.2 Generalization to non-regular orbits

The aim of this subsection is to generalize the results of the last subsection to non-regular semisimple coadjoint orbits. To achieve this, we need to replace $\mathfrak{h}$ by a possibly larger stabilizer $\mathfrak{g}_{\lambda}$ and define a generalization of the Shapovalov pairing. When this pairing is non-degenerate, we derive an explicit formula for its canonical element, which satisfies (2.16).

Let $\mathfrak{g}$ be a complex semisimple Lie algebra acting under the coadjoint action, i.e. the action dual to the adjoint action, on its dual $\mathfrak{g}^{*}$. We assume that $\lambda\in\mathfrak{g}^{*}$ is semisimple (as defined in Subsection 1.1) with stabilizer $\mathfrak{g}_{\lambda}\coloneqq\{X\in\mathfrak{g}\mid\operatorname{\mathrm{ad}}_{X}^{*}\lambda=0\}$. We fix a Cartan subalgebra $\mathfrak{h}$ containing $\lambda^{\sharp}$ (which is possible since $\lambda$ is semisimple) and denote the corresponding root system by $\Delta$. Since any $H\in\mathfrak{h}$ commutes with $\lambda^{\sharp}$, it follows that $\operatorname{\mathrm{ad}}_{H}^{*}\lambda=\lambda([-H,{}\cdot{}])=-B(\lambda^{\sharp},[H,{}\cdot{}])=-B([\lambda^{\sharp},H],{}\cdot{})=0$, so $\mathfrak{h}\subseteq\mathfrak{g}_{\lambda}$. We let

[TABLE]

One checks easily that $\mathfrak{g}_{\lambda}=\mathfrak{h}\oplus\bigoplus_{\alpha\in\Delta^{\prime}}\mathfrak{g}^{\alpha}$. Given an ordering on $\Delta$ with $\Delta^{\pm}$ being the set of positive respectively negative roots, define $\smash{\hat{\Delta}^{\pm}}=\Delta^{\pm}\cap\smash{\hat{\Delta}}$ and $(\Delta^{\prime})^{\pm}=\Delta^{\pm}\cap\Delta^{\prime}$. Furthermore, let ${\tilde{\mathfrak{n}}}^{\pm}\coloneqq\bigoplus_{\alpha\in\hat{\Delta}^{\pm}}\mathfrak{g}^{\alpha}$ and ${\tilde{\mathfrak{b}}}^{\pm}\coloneqq\mathfrak{g}_{\lambda}\oplus{\tilde{\mathfrak{n}}}^{\pm}$.

Definition \thedefinition.

*An ordering of $\Delta$ is called invariant if for any $\alpha\in\hat{\Delta}^{+}$ and $\beta\in\Delta^{\prime}$ such that $\alpha+\beta$ is again a root, this root $\alpha+\beta$ is in $\hat{\Delta}^{+}$. *

Note that since the sum of two roots in $\Delta^{\prime}$ is again in $\Delta^{\prime}$ (if it is a root), it is automatic that $\alpha+\beta\in\hat{\Delta}$. The important part of the previous definition is that $\alpha+\beta$ should again be positive. See Figure 3 for an example of invariant and non-invariant orderings.

Lemma 2.7**.**

*An ordering of $\Delta$ is invariant if and only if $\alpha+\beta\in\hat{\Delta}^{+}$ holds for any $\alpha,\beta\in\hat{\Delta}^{+}$ with $\alpha+\beta\in\Delta$. *

In the condition of the lemma it is automatic that $\alpha+\beta$ is positive and the important part is that it lies in $\hat{\Delta}$.

Proof**:**

Assume the condition of the lemma is false, i.e. $\alpha,\beta\in\hat{\Delta}^{+}$ and $\alpha+\beta\in\Delta\setminus\hat{\Delta}^{+}$. Since $\alpha+\beta$ is positive we must then have $\alpha+\beta\in\Delta^{\prime}$. Consequently $\alpha+(-(\alpha+\beta))=-\beta\notin\hat{\Delta}^{+}$, so the ordering is not invariant.

Conversely, if the ordering is not invariant, then we can find $\alpha\in\hat{\Delta}^{+}$ and $\beta\in\Delta^{\prime}$ such that $\alpha+\beta\in\Delta\setminus\hat{\Delta}^{+}$. Then we must have $\alpha+\beta\in\hat{\Delta}^{-}$ and therefore $\alpha+(-(\alpha+\beta))=-\beta\notin\hat{\Delta}^{+}$, so the condition of the lemma is not fulfilled. $\boxempty$

Intuitively the invariance of an ordering means that roots in $\Delta^{\prime}$ are close to being simple, or more precisely that they are linear combinations of simple roots in $\Delta^{\prime}$. Indeed, if $\alpha\in(\Delta^{\prime})^{+}$, then $\alpha$ is a non-negative linear combination of simple roots. By the lemma at least one of those simple roots, say $\sigma$, must be in $\Delta^{\prime}$, so $\alpha=\sigma$ or $\alpha-\sigma\in(\Delta^{\prime})^{+}$ and we can apply induction.

Corollary \thecorollary.

*If the ordering of $\Delta$ is invariant, then ${\tilde{\mathfrak{n}}}^{\pm}$ and ${\tilde{\mathfrak{b}}}^{\pm}$ are both Lie subalgebras of $\mathfrak{g}$. Moreover, $[\mathfrak{g}_{\lambda},{\tilde{\mathfrak{n}}}^{\pm}]\subseteq{\tilde{\mathfrak{n}}}^{\pm}$ and $[\mathfrak{g}_{\lambda},{\tilde{\mathfrak{b}}}^{\pm}]\subseteq{\tilde{\mathfrak{b}}}^{\pm}$. *

Proof**:**

The condition in the previous lemma says precisely that $[{\tilde{\mathfrak{n}}}^{\pm},{\tilde{\mathfrak{n}}}^{\pm}]\subseteq{\tilde{\mathfrak{n}}}^{\pm}$, i.e. that ${\tilde{\mathfrak{n}}}^{\pm}$ is a Lie subalgebra of $\mathfrak{g}$. The defining property of an invariant ordering means that $[\mathfrak{g}_{\lambda},{\tilde{\mathfrak{n}}}^{\pm}]\subseteq{\tilde{\mathfrak{n}}}^{\pm}$. The statements for ${\tilde{\mathfrak{b}}}^{\pm}$ are then clear. $\boxempty$

Definition \thedefinition.

*We say an ordering is standard if there is a set $S\subseteq\mathbbm{C}\setminus\{0\}$, closed under addition and satisfying $S\cap(-S)=\emptyset$, $S\cup(-S)=\mathbbm{C}\setminus\{0\}$ such that $\alpha\in\hat{\Delta}$ is positive if and only if $(\alpha,\lambda)\in S$. *

Standard invariant orderings exist always since we can construct them as follows. First, take any ordering on the set $\Delta^{\prime}$ (meaning a subset $(\Delta^{\prime})^{+}$ such that if the sum of two elements of $(\Delta^{\prime})^{+}$ is in $\Delta^{\prime}$, then it is in $(\Delta^{\prime})^{+}$ and such that for $(\Delta^{\prime})^{-}\coloneqq-(\Delta^{\prime})^{+}$ we have $(\Delta^{\prime})^{+}\cup(\Delta^{\prime})^{-}=\Delta^{\prime}$ and $(\Delta^{\prime})^{+}\cap(\Delta^{\prime})^{-}=\emptyset$). Then choose a set $S$ that is closed under addition and satisfies $S\cap(-S)=\emptyset$ and $S\cup(-S)=\mathbbm{C}\setminus\{0\}$, e.g. S=\{z\in\mathbbm{C}\setminus\{0\}\mid\mathsf{Re}(z)>0\text{ or z\in\mathrm{i}\mathbbm{R}^{+}}\}. Let $\alpha\in\Delta$ be positive if $\alpha\in(\Delta^{\prime})^{+}$ or $(\alpha,\lambda)\in S$.

For real coadjoint orbits standard invariant orderings are the ones which induce star products of pseudo Wick type (under some further assumptions, see Subsection 4.3), and therefore the orderings we are mainly interested in. However, the construction below works also for other (possibly non-standard) invariant orderings.

Before generalizing the results of the last subsection, we would like to mention the following technical lemma for later use:

Lemma 2.8**.**

*Let $\mathfrak{g}$ be a semisimple Lie algebra, let $\lambda\in\mathfrak{g}^{*}$ be semisimple, and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$ containing $\lambda^{\sharp}$. Assume that we have chosen an invariant ordering defining sets $\Delta^{+}$, $\smash{\hat{\Delta}}$, and $\Delta^{\prime}$ as above. Then there is a constant $M\in\mathbbm{N}$ such that for any $m\in\mathbbm{N}$ the sum of $m$ positive roots in $\hat{\Delta}^{+}$ and at least $Mm$ positive roots in $(\Delta^{\prime})^{+}$ is not in $\mathbbm{N}_{0}\hat{\Delta}^{+}$. *

Proof**:**

Label the simple roots by $\sigma_{1},\dots,\sigma_{r}$ such that the first $r^{\prime}$ simple roots $\sigma_{1},\dots,\sigma_{r^{\prime}}$ are in $\Delta^{\prime}$ and the remaining simple roots are in $\hat{\Delta}$. Label all roots in $\hat{\Delta}^{+}$ by $\alpha_{1},\dots,\smash{\alpha_{\tilde{k}}}$. Then there are unique non-negative integers $c^{i}_{j}\in\mathbbm{N}_{0}$ such that $\alpha_{j}=\sum_{i=1}^{r}c^{i}_{j}\sigma_{i}$. Set $M^{\prime}=\smash{\max_{j\in\{1,\dots,\tilde{k}\}}\sum_{i=1}^{r^{\prime}}c^{i}_{j}}$, $M^{\prime\prime}=\max_{j\in\{1,\dots,\tilde{k}\}}\sum_{i=r^{\prime}+1}^{r}c^{i}_{j}$ and $M=M^{\prime}M^{\prime\prime}+1$.

Since $\alpha_{j}\in\smash{\hat{\Delta}^{+}}$ we have $\sum_{i=r^{\prime}+1}^{r}c_{j}^{i}\geq 1$, and $\sum_{i=1}^{r^{\prime}}c^{i}_{j}\leq M^{\prime}\leq M^{\prime}\sum_{i=r^{\prime}+1}^{r}c^{i}_{j}$ for any $j\in\{1,\dots,\tilde{k}\}$. Note that any element $\beta\in\mathbbm{N}_{0}\hat{\Delta}^{+}$ can be written uniquely as $\beta=\sum_{i=1}^{r}\beta^{i}\sigma_{i}$ with $\beta^{i}\in\mathbbm{N}_{0}$, and the coefficients satisfy the same inequality $\sum_{i=1}^{r^{\prime}}\beta^{i}\leq M^{\prime}\sum_{i=r^{\prime}+1}^{r}\beta^{i}$.

Recall that any root in $(\Delta^{\prime})^{+}$ is a linear combination of simple roots in $(\Delta^{\prime})^{+}$. So if $\sum_{i=1}^{r}d^{i}\sigma_{i}\in(\Delta^{\prime})^{+}$, then $d^{i}=0$ for all $i=r^{\prime}+1,\dots,r$. Therefore, if $\gamma$ is the sum of $m$ roots from $\hat{\Delta}^{+}$ and at least $Mm$ roots from $(\Delta^{\prime})^{+}$, and $\gamma=\sum_{i=1}^{r}\gamma^{i}\sigma_{i}$, then $M^{\prime}\sum_{i=r^{\prime}+1}^{r}\gamma^{i}\leq M^{\prime}M^{\prime\prime}m<Mm\leq\sum_{i=1}^{r^{\prime}}\gamma^{i}$, so $\gamma$ cannot be in $\mathbbm{N}_{0}\hat{\Delta}^{+}$. $\boxempty$

Note that for a regular coadjoint orbit, we have $\Delta^{\prime}=\emptyset$. Consequently $\hat{\Delta}=\Delta$, $\mathfrak{g}_{\lambda}=\mathfrak{h}$, ${\tilde{\mathfrak{n}}}^{+}={\mathfrak{n}}^{+}$ and ${\tilde{\mathfrak{n}}}^{-}={\mathfrak{n}}^{-}$. In this case every ordering is invariant, and the generalized Shapovalov pairing, that we will introduce now, coincides with the Shapovalov pairing introduced in the last subsection. Since $\mathfrak{g}_{\lambda}=\mathfrak{h}$ when $\Delta^{\prime}=\emptyset$, we usually denote an element of $\mathfrak{g}_{\lambda}$ by $H$.

Let $\lambda\in\mathfrak{g}_{\lambda}^{*}$ be the restriction of $\lambda\in\mathfrak{g}^{*}$ to $\mathfrak{g}_{\lambda}$. Then $\lambda([H^{\prime},H])=\operatorname{\mathrm{ad}}^{*}_{H}\lambda(H^{\prime})=0$ for all $H,H^{\prime}\in\mathfrak{g}_{\lambda}$, so $H\mathbin{\triangleright}z=\lambda(H)z$ makes $\mathbbm{C}$ a left or right $\mathfrak{g}_{\lambda}$-module. Extending trivially along ${\tilde{\mathfrak{n}}}^{\pm}$ gives a left or right ${\tilde{\mathfrak{b}}}^{\pm}$-module, and we denote the corresponding left $\mathscr{U}({\tilde{\mathfrak{b}}}^{\pm})$-module by $\tilde{\mathbbm{C}}^{\pm}_{\lambda}$ and the right $\mathscr{U}({\tilde{\mathfrak{b}}}^{-})$-module by $\tilde{\mathbbm{C}}^{*}_{\lambda}$. Define the generalized Verma modules

[TABLE]

$\tilde{M}_{\lambda}$ and $\tilde{M}_{\lambda}^{-}$ are left $\mathscr{U}\mathfrak{g}$-modules, $\tilde{M}_{\lambda}^{*}$ is a right $\mathscr{U}\mathfrak{g}$-module. Most of the results of the previous subsection have obvious analogues in this setting.

Let $\{X_{1},\dots,X_{\tilde{k}}\}$ be a basis of ${\tilde{\mathfrak{n}}}^{+}$, $\{Y_{1},\dots,Y_{\tilde{k}}\}$ be a basis of ${\tilde{\mathfrak{n}}}^{-}$, and $\{H_{1},\dots,H_{\tilde{r}}\}$ be a basis of $\mathfrak{g}_{\lambda}$. Since $\mathfrak{g}={\tilde{\mathfrak{n}}}^{+}\oplus\mathfrak{g}_{\lambda}\oplus{\tilde{\mathfrak{n}}}^{-}$ the Poincaré–Birkhoff–Witt theorem implies that

[TABLE]

are bases for $\mathscr{U}\mathfrak{g}$. Define maps

[TABLE]

Note that they are compatible with the maps $\pi_{\lambda}^{-}$, $\pi_{\lambda}^{+}$, and $\pi_{\lambda}^{*}$ in the sense that $\tilde{\pi}_{\lambda}^{-}\circ\pi_{\lambda}^{-}=\tilde{\pi}_{\lambda}^{-}$, $\tilde{\pi}_{\lambda}^{+}\circ\pi_{\lambda}^{+}=\tilde{\pi}_{\lambda}^{+}$, and $\tilde{\pi}_{\lambda}^{*}\circ\pi_{\lambda}^{*}=\tilde{\pi}_{\lambda}^{*}$. On the left hand sides, we are implicitly using the inclusion $\mathscr{U}({\mathfrak{n}}^{\pm})\to\mathscr{U}\mathfrak{g}$. Note that this inclusion is not a $\mathscr{U}\mathfrak{g}$-module map.

Lemma 2.9**.**

The maps ${}\cdot{}\mathbin{\otimes}1\colon\mathscr{U}({\tilde{\mathfrak{n}}}^{-})\to\tilde{M}_{\lambda}$, $v\mapsto v\mathbin{\otimes}1$ and ${}\cdot{}\mathbin{\otimes}1\colon\mathscr{U}({\tilde{\mathfrak{n}}}^{+})\to\tilde{M}^{-}_{\lambda}$, $u\mapsto u\mathbin{\otimes}1$ define isomorphisms of left $\mathscr{U}({\tilde{\mathfrak{n}}}^{-})$-modules and $\mathscr{U}({\tilde{\mathfrak{n}}}^{+})$-modules, respectively. The map $1\mathbin{\otimes}{}\cdot{}\colon\mathscr{U}({\tilde{\mathfrak{n}}}^{+})\to M^{*}_{\lambda}$, $u\mapsto 1\mathbin{\otimes}u$ is an isomorphism of right $\mathscr{U}({\tilde{\mathfrak{n}}}^{+})$-modules. The $\mathscr{U}\mathfrak{g}$-module structures on $\mathscr{U}({\tilde{\mathfrak{n}}}^{\pm})$ obtained by transferring the module structures on the generalized Verma modules with these isomorphisms are given explicitly by

[TABLE]

*Furthermore, $S(w\mathbin{\tilde{\mathbin{\triangleright}}_{\lambda}^{+}}u)=S(u)\mathbin{\tilde{\mathbin{\triangleleft}}_{\lambda}^{*}}S(w)$, where $S$ denotes the antipode of $\mathscr{U}\mathfrak{g}$. *

Proof**:**

Similar to the proof of Lemma 2.1. $\boxempty$

Note that since $\mathscr{U}({\tilde{\mathfrak{n}}}^{\pm})$ is a $\mathscr{U}\mathfrak{g}$-module, we must have

[TABLE]

and

[TABLE]

for all $w,w^{\prime}\in\mathscr{U}\mathfrak{g}$. In particular, this implies that the map \smash{\tilde{\pi}_{\lambda}^{\pm}\big{|}_{\mathscr{U}({\mathfrak{n}}^{\pm})}}\colon\mathscr{U}({\mathfrak{n}}^{\pm})\to\mathscr{U}({\tilde{\mathfrak{n}}}^{\pm}) is a $\mathscr{U}\mathfrak{g}$-module homomorphism (with respect to the module structures given by $\mathbin{\mathbin{\triangleright}_{\lambda}^{\pm}}$ and $\mathbin{\tilde{\mathbin{\triangleright}}_{\lambda}^{\pm}}$). Indeed, for the plus case we have

[TABLE]

for all $w\in\mathscr{U}\mathfrak{g}$ and $u\in\mathscr{U}({\mathfrak{n}}^{+})$ and the minus case is similar. Define $\mathfrak{g}_{\lambda}^{\pm}\coloneqq\bigoplus_{\alpha\in(\Delta^{\prime})^{\pm}}\mathfrak{g}^{\alpha}=\mathfrak{g}_{\lambda}\cap\mathfrak{n}^{\pm}$. Note that $\mathscr{U}\mathfrak{g}\cdot\mathfrak{g}_{\lambda}^{\pm}=\{w\mathbin{\mathbin{\triangleright}_{\lambda}^{\pm}}X\mid w\in\mathscr{U}\mathfrak{g},X\in\mathfrak{g}_{\lambda}^{\pm}\}$ is a $\mathscr{U}\mathfrak{g}$-submodule of $\mathscr{U}({\mathfrak{n}}^{\pm})$. Since $\tilde{\pi}_{\lambda}^{\pm}$ is a map of $\mathscr{U}\mathfrak{g}$-modules and vanishes on $\mathfrak{g}_{\lambda}^{\pm}$, $\mathscr{U}\mathfrak{g}\cdot\mathfrak{g}_{\lambda}^{\pm}$ is in its kernel.

Lemma 2.10**.**

*The induced maps $\tilde{\pi}^{\pm}_{\lambda}\colon\mathscr{U}({\mathfrak{n}}^{\pm})/\mathscr{U}\mathfrak{g}\cdot\mathfrak{g}_{\lambda}^{\pm}\to\mathscr{U}({\tilde{\mathfrak{n}}}^{\pm})$ are isomorphisms of $\mathscr{U}\mathfrak{g}$-modules. *

Proof**:**

It is easy to check that the quotient map induced by the inclusion $\mathscr{U}({\tilde{\mathfrak{n}}}^{\pm})\to\mathscr{U}({\mathfrak{n}}^{\pm})$ defines an inverse. $\boxempty$

As before there are isomorphisms $\tilde{M}_{\lambda}^{*}\mathbin{\otimes}_{\mathscr{U}\mathfrak{g}}\tilde{M}_{\lambda}\cong\tilde{\mathbbm{C}}^{-}_{\lambda}\mathbin{\otimes}_{\mathscr{U}({\tilde{\mathfrak{b}}}^{-})}\mathscr{U}\mathfrak{g}\mathbin{\otimes}_{\mathscr{U}({\tilde{\mathfrak{b}}}^{+})}\tilde{\mathbbm{C}}_{\lambda}\cong\tilde{\mathbbm{C}}^{-}_{\lambda}\mathbin{\otimes}_{\mathscr{U}(\mathfrak{g}_{\lambda})}\tilde{\mathbbm{C}}_{\lambda}\cong\mathbbm{C}$, which we use to define the Shapovalov pairings ${\langle{}\cdot{},{}\cdot{}\rangle^{\sim}_{\lambda}}^{\prime}\colon\tilde{M}_{\lambda}^{*}\times\tilde{M}_{\lambda}\to\mathbbm{C}$, $(x,y)\mapsto{\langle x,y\rangle^{\sim}_{\lambda}}^{\prime}\coloneqq x\mathbin{\otimes}y$ and

[TABLE]

In the same way as in Lemma 2.2 one proves that this pairing can be computed by

[TABLE]

Note that $\tilde{\pi}^{-}_{\lambda}\circ\tilde{\pi}_{\lambda}^{*}=\tilde{\pi}_{\lambda}^{*}\circ\tilde{\pi}_{\lambda}^{-}=\pi_{\lambda}^{*}\circ\pi_{\lambda}^{-}=\pi_{\lambda}$, so there is no need to introduce a $\tilde{\pi}_{\lambda}$.

Lemma 2.11**.**

*Let $u\in\mathscr{U}({\mathfrak{n}}^{+})$ and $v\in\mathscr{U}({\mathfrak{n}}^{-})$. Then we have $\langle\tilde{\pi}_{\lambda}^{+}u,\tilde{\pi}_{\lambda}v\rangle^{\sim}_{\lambda}=\langle u,v\rangle_{\lambda}$. In particular \langle{}\cdot{},{}\cdot{}\rangle_{\lambda}\big{|}_{\mathscr{U}({\mathfrak{n}}^{+})\times\mathscr{U}\mathfrak{g}\cdot\mathfrak{g}_{\lambda}^{-}}=\langle{}\cdot{},{}\cdot{}\rangle_{\lambda}\big{|}_{\mathscr{U}\mathfrak{g}\cdot\mathfrak{g}_{\lambda}^{+}\times\mathscr{U}({\mathfrak{n}}^{-})}=0. *

Proof**:**

Using (2.22) twice, we compute

[TABLE]

$\boxempty$

Define the set

[TABLE]

Furthermore, let $\tilde{W}$ be the set of words $w\in W$ such that $\alpha_{w_{i\dots\lvert w\rvert}}\in\mathbbm{N}_{0}\hat{\Delta}^{+}$ for all $i=1,\dots,\lvert w\rvert$. Since $\tilde{\pi}_{\lambda}^{+}(X_{w})=0$ and $\tilde{\pi}_{\lambda}^{-}(Y_{w})=0$ for $w\in W\setminus\tilde{W}$, the following theorem is not surprising.

Theorem \thethm.

Let $\lambda\in\tilde{\Lambda}$. Then the Shapovalov pairing $\langle{}\cdot{},{}\cdot{}\rangle^{\sim}_{\lambda}\colon\mathscr{U}({\tilde{\mathfrak{n}}}^{+})\times\mathscr{U}({\tilde{\mathfrak{n}}}^{-})\to\mathbbm{C}$ is non-degenerate and its canonical element $F_{\lambda}\in\mathscr{U}({\tilde{\mathfrak{n}}}^{+})\operatorname{\hat{\otimes}}\mathscr{U}({\tilde{\mathfrak{n}}}^{-})$ is given by

[TABLE]

Proof**:**

It suffices to prove that $\sum_{w\in\tilde{W}}p^{w}_{\lambda}(\alpha_{w})^{-1}\tilde{\pi}_{\lambda}^{-}(Y_{w})\langle\tilde{\pi}_{\lambda}^{+}(X_{w}),\tilde{v}\rangle^{\sim}_{\lambda}=\tilde{v}$ for all $\tilde{v}\in\mathscr{U}({\tilde{\mathfrak{n}}}^{-})$ and that $\sum_{w\in\tilde{W}}p^{w}_{\lambda}(\alpha_{w})^{-1}\tilde{\pi}_{\lambda}^{+}(X_{w})\langle\tilde{u},\tilde{\pi}_{\lambda}^{-}(Y_{w})\rangle^{\sim}_{\lambda}=\tilde{u}$ for all $\tilde{u}\in\mathscr{U}({\tilde{\mathfrak{n}}}^{+})$ by using an analogue of Lemma 2.3. Let $v\in\mathscr{U}({\mathfrak{n}}^{-})$ be the image of $\tilde{v}$ under the inclusion $\mathscr{U}({\tilde{\mathfrak{n}}}^{-})\to\mathscr{U}({\mathfrak{n}}^{-})$, so that $\tilde{\pi}_{\lambda}^{-}(v)=\tilde{v}$. Assume that $v=\sum_{\mu\in\mathbbm{N}_{0}\hat{\Delta}^{+}}v_{-\mu}$ is the weight decomposition of $v$. Then

[TABLE]

where $\tilde{W}_{\mu}=\{w\in\tilde{W}\mid\alpha_{w}=\mu\}$. We claim that there is an admissible tree $T$ and $v^{\prime}\in\mathscr{U}\mathfrak{g}\cdot\mathfrak{g}_{\lambda}^{-}$ such that

[TABLE]

which would finish the proof by using Lemma 2.5. Indeed, let

[TABLE]

be the smallest tree containing $\tilde{W}_{\mu}$. Since $\lambda\in\tilde{\Lambda}$, this tree is admissible. Furthermore $\tilde{W}_{\mu}\subseteq\max T$ and any element $w\in\max T$ satisfies exactly one of the following two conditions. Either $\alpha_{w}=\mu$, so that $w\in\tilde{W}_{\mu}$ appears in the sum on the left hand side of the above equation. Or $\mu-\alpha_{w}\notin\mathbbm{N}_{0}\hat{\Delta}^{+}$, so that $X_{w^{\mathrm{opp}}}v_{-\mu}$ would have to be of weight $\alpha_{w}-\mu\notin-\mathbbm{N}_{0}\hat{\Delta}^{+}$ and does therefore either vanish or lie in $\mathscr{U}\mathfrak{g}\cdot\mathfrak{g}_{\lambda}^{-}$. The statement for $\tilde{u}$ is proven similarly. $\boxempty$

Using the inclusions $\mathscr{U}({\tilde{\mathfrak{n}}}^{\pm})\to\mathscr{U}\mathfrak{g}$ and the projection $\mathscr{U}\mathfrak{g}\to\mathscr{U}\mathfrak{g}/\mathscr{U}\mathfrak{g}\cdot\mathfrak{g}_{\lambda}$, we map $F_{\lambda}$ to $\smash{(\mathscr{U}\mathfrak{g}/\mathscr{U}\mathfrak{g}\cdot\mathfrak{g}_{\lambda})^{\hat{\mathbin{\otimes}}2}}$. Note that, as before, $\mathscr{U}\mathfrak{g}\cdot\mathfrak{g}_{\lambda}$ is a homogeneous ideal in $\mathscr{U}\mathfrak{g}$, so the grading of $\mathscr{U}\mathfrak{g}$ stays well-defined on the quotient. The action of $\mathfrak{g}_{\lambda}$ on $(\mathscr{U}\mathfrak{g})^{\mathbin{\otimes}2}$ also passes to the quotient and extends to a continuous action on the completed tensor product.

Theorem \thethm (Alekseev–Lachowska [1]).

Let $\lambda\in\tilde{\Lambda}$. Then $F_{\lambda}\in(\mathscr{U}\mathfrak{g}/\mathscr{U}\mathfrak{g}\cdot\mathfrak{g}_{\lambda})^{\hat{\mathbin{\otimes}}2}$ is $\mathfrak{g}_{\lambda}$-invariant and satisfies

[TABLE]

*in $(\mathscr{U}\mathfrak{g}/\mathscr{U}\mathfrak{g}\cdot\mathfrak{g}_{\lambda})^{\hat{\mathbin{\otimes}}3}$. *

Proof**:**

Note that the $\mathfrak{g}$-invariance of the Shapovalov pairing (proven similarly as in Lemma 2.2) implies that $F_{\lambda}\in\mathscr{U}({\tilde{\mathfrak{n}}}^{+})\operatorname{\hat{\otimes}}\mathscr{U}({\tilde{\mathfrak{n}}}^{-})$ is also $\mathfrak{g}$-invariant. Then $F_{\lambda}\in\smash{(\mathscr{U}\mathfrak{g}/\mathscr{U}\mathfrak{g}\cdot\mathfrak{g}_{\lambda})^{\hat{\mathbin{\otimes}}2}}$ is $\mathfrak{g}_{\lambda}$-invariant since the map $\mathscr{U}({\tilde{\mathfrak{n}}}^{+})\times\mathscr{U}({\tilde{\mathfrak{n}}}^{-})\to\smash{(\mathscr{U}\mathfrak{g}/\mathscr{U}\mathfrak{g}\cdot\mathfrak{g}_{\lambda})^{\hat{\mathbin{\otimes}}2}}$ is $\mathfrak{g}_{\lambda}$-equivariant. Equation (2.28) is proven in [1, Section 4]. $\boxempty$

It will be convenient in the following to write $F_{\lambda}$ as a sum of elements that are all invariant under $\mathfrak{g}_{\lambda}$.

Lemma 2.12**.**

Let $\lambda\in\tilde{\Lambda}$. Then there is a partition of $\tilde{W}$ into finite subsets $\tilde{W}_{\ell}$, $\ell\in\mathbbm{N}_{0}$ such that

[TABLE]

*is $\mathfrak{g}_{\lambda}$-invariant. *

Proof**:**

It will be convenient to introduce a different grading $d^{\prime}$ on $\mathfrak{g}$, for which $\mathfrak{g}_{\lambda}$ is of degree [math]. To this end, let $\mathfrak{h}$ and the root spaces of simple roots in $\Delta^{\prime}$ be of degree [math], and let the root spaces of simple roots in $\hat{\Delta}$ be of degree $1$. Since any root is a unique linear combination of simple roots this assignment extends to a grading on $\mathfrak{g}$. More explicitly, if $\sigma_{1},\dots,\sigma_{r}\in\Delta$ are the simple roots, with $\sigma_{1},\dots,\sigma_{r^{\prime}}\in\Delta^{\prime}$, then the root space of a root $\alpha=\sum_{i=1}^{r}c^{i}\sigma_{i}$ is of degree $d^{\prime}(\alpha)=\sum_{i=r^{\prime}+1}^{r}c^{i}$. Since $\mathfrak{g}_{\lambda}$ is spanned by $\mathfrak{h}$ and the root spaces of roots in $\Delta^{\prime}$, and since the invariance of the ordering implies that any root in $\Delta^{\prime}$ is a linear combination of simple roots in $\Delta^{\prime}$, it follows that every element of $\mathfrak{g}_{\lambda}$ is homogeneous of degree [math]. This grading is coarser than the grading given by $d$, in the sense that the graded components with respect to the new grading $d^{\prime}$ are direct sums of the graded components with respect to $d$. The restrictions of the maps $\tilde{\pi}_{\lambda}^{\pm}$ to $\mathscr{U}({\mathfrak{n}}^{\pm})$ are homogeneous of degree [math] with respect to (the restriction of) the $\mathbbm{Z}$-grading on $\mathscr{U}\mathfrak{g}$ induced by $d^{\prime}$.

For $w\in W$ set $d^{\prime}(w)\coloneqq d^{\prime}(\alpha_{w_{1}})+\dots+d^{\prime}(\alpha_{w_{\lvert w\rvert}})$, and define $\smash{\tilde{W}_{\ell}}\coloneqq\{w\in\smash{\tilde{W}}\mid d^{\prime}(w)=\ell\}$. It follows from Lemma 2.8 that $\smash{\tilde{W}_{\ell}}$ is finite for every $\ell$. The elements $F_{\lambda,\ell}$ defined from $\smash{\tilde{W}}_{\ell}$ as in (2.29) have a nice description in terms of the grading $d^{\prime}$. Since all graded components of ${\tilde{\mathfrak{n}}}^{+}$ resp. ${\tilde{\mathfrak{n}}}^{-}$ are of degree $\geq 1$ resp. $\leq-1$, $d^{\prime}$ induces a grading of $\mathscr{U}({\tilde{\mathfrak{n}}}^{+})\mathbin{\otimes}\mathscr{U}({\tilde{\mathfrak{n}}}^{-})$ by $\mathbbm{N}_{0}\times(-\mathbbm{N}_{0})$. Using the homogeneity of $\tilde{\pi}_{\lambda}^{\pm}$, it follows directly from the definition of $\smash{\tilde{W}_{\ell}}$ that $F_{\lambda,\ell}$ is precisely the component of $F_{\lambda}$ of degree $(\ell,-\ell)$ with respect to this grading. Since $\mathfrak{g}_{\lambda}$ is of degree [math], the action of $\mathfrak{g}_{\lambda}$ on $\mathscr{U}({\tilde{\mathfrak{n}}}^{+})\mathbin{\otimes}\mathscr{U}({\tilde{\mathfrak{n}}}^{-})$ preserves the graded components, and the $\mathfrak{g}_{\lambda}$-invariance of $F_{\lambda}$ implies that all the graded components $F_{\lambda,\ell}$ must also be $\mathfrak{g}_{\lambda}$-invariant. $\boxempty$

2.3 The induced formal and strict products

In this subsection we construct associative products from the element $F_{\lambda}$ obtained at the end of the last subsection. We will rescale $\lambda$ in order to introduce a parameter playing the role of Planck’s constant in the construction. Then we would like to use the results of Subsection 1.2 to obtain bidifferential operators from (the rescaled) $F_{\lambda}$. However, since $F_{\lambda}$ is only in the completed tensor product, applying these results naively would give a sum of bidifferential operators of increasing orders and we have to deal with its convergence. There are essentially two solutions to this problem: Firstly, we can take a formal expansion in the parameter $\hbar$, which will give us a well-defined power series of bidifferential operators of increasing order. Secondly, we can restrict ourselves to applying these operators only to some polynomial functions, for which only finitely many terms of the infinite sum give a non-zero contribution. We discuss both approaches in detail, starting with the formal one.

Let us first introduce the rescaling. Define the set

[TABLE]

and for $\hbar\in\mathbbm{C}\setminus P_{\lambda}$ set $F_{\hbar}\coloneqq F_{\mathrm{i}\lambda/\hbar}$ and $F_{\hbar,\ell}\coloneqq F_{\mathrm{i}\lambda/\hbar,\ell}$, where $F_{\mathrm{i}\lambda/\hbar}$ was computed in Subsection 2.2 and $F_{\mathrm{i}\lambda/\hbar,\ell}$ was defined in Lemma 2.12. Note that $\mathfrak{g}_{\mathrm{i}\lambda/\hbar}=\mathfrak{g}_{\lambda}$, so $F_{\hbar}\in\smash{((\mathscr{U}\mathfrak{g}/\mathscr{U}\mathfrak{g}\cdot\mathfrak{g}_{\lambda})^{\operatorname{\hat{\otimes}}2})^{\mathfrak{g}_{\lambda}}}$ holds for all $\hbar\in\mathbbm{C}\setminus P_{\lambda}$. Furthermore, the projections \smash{\tilde{\pi}_{\mathrm{i}\lambda/\hbar}^{\pm}\big{|}_{\mathscr{U}({\mathfrak{n}}^{\pm})}}\colon\mathscr{U}({\mathfrak{n}}^{\pm})\to\mathscr{U}({\tilde{\mathfrak{n}}}^{\pm}) are independent of $\hbar$, which one can easily see from their definition in (2.20).

Proposition \theproposition.

*Let $\mathfrak{g}$ be a complex semisimple Lie algebra, $\mathfrak{h}$ a Cartan subalgebra of $\mathfrak{g}$, and $\lambda\in\mathfrak{h}^{*}$. Fix an invariant ordering on $\Delta$, and assume that $(\lambda,\mu)\neq 0$ for all $\mu\in\mathbbm{N}_{0}\hat{\Delta}^{+}$ satisfying $\frac{1}{2}(\mu,\mu)=(\rho,\mu)$. Then the set $P_{\lambda}$ is countable and accumulates only at zero. *

Proof**:**

From the definition of $P_{\lambda}$ we obtain

[TABLE]

Under our assumptions the function $\hbar\mapsto p_{\mathrm{i}\lambda/\hbar}(\mu)=\frac{1}{2}(\mu,\mu)-(\rho,\mu)-\frac{\mathrm{i}}{\hbar}(\lambda,\mu)$ has the only root $\mathrm{i}(\lambda,\mu)/(\frac{1}{2}(\mu,\mu)-(\rho,\mu))$ if $\frac{1}{2}(\mu,\mu)-(\rho,\mu)\neq 0$ and no root otherwise. Therefore $P_{\lambda}$ is countable since $\mathbbm{N}_{0}\hat{\Delta}^{+}\setminus\{0\}$ is countable. Furthermore, $P_{\lambda}$ accumulates only at zero since

[TABLE]

if $\lVert\mu\rVert>2\lVert\rho\rVert$. Note that there are only finitely many elements $\mu\in\mathbbm{N}_{0}\hat{\Delta}^{+}$ with $\lVert\mu\rVert\leq 2\lVert\rho\rVert$. $\boxempty$

Remark \theremark.

If the ordering in the previous proposition is standard, then any element $\mu\in\mathbbm{N}_{0}\hat{\Delta}^{+}$ automatically satisfies $(\lambda,\mu)\neq 0$: For all $\alpha\in\hat{\Delta}^{+}$ we have $(\lambda,\alpha)\in S$ and since $S$ is closed under addition this implies $(\lambda,\mu)\in S$ for all $\mu\in\mathbbm{N}_{0}\hat{\Delta}^{+}$. Note that $0\notin S$, so in particular $(\lambda,\mu)\neq 0$.

Note also that $\frac{1}{2}(\mu,\mu)=(\rho,\mu)$ implies $\lVert\mu\rVert\leq 2\lVert\rho\rVert$, so there can only be finitely many elements $\mu\in\mathbbm{N}_{0}\Delta$ satisfiying $\frac{1}{2}(\mu,\mu)=(\rho,\mu)$. Among those are all simple roots and the element $2\rho$. However, simple roots which are in $\smash{\mathbbm{N}_{0}\hat{\Delta}}$ are by definition not orthogonal to $\lambda$. An example of an element that is not a simple root and not $2\rho$ in the case of $\mathfrak{g}=\mathfrak{sl}_{3}(\mathbbm{C})$ with root system as in Figure 1 is $\mu=\alpha_{1}+\alpha_{2}$.

We say that $F_{\hbar}$ depends rationally on $\hbar$ if all the $F_{\hbar,\ell}$ depend rationally on $\hbar$. This makes sense since $F_{\hbar,\ell}$ takes values in a finite dimensional subspace of $(\mathscr{U}\mathfrak{g}/\mathscr{U}\mathfrak{g}\cdot\mathfrak{g}_{\lambda})^{\mathbin{\otimes}2}$ that is independent of $\hbar$.

**Theorem \thethm (Alekseev–Lachowska

[1]).**

*Let $\lambda\in\mathfrak{h}^{*}$ and assume that $P_{\lambda}$ is countable. Then $F_{\hbar}$ depends rationally on $\hbar$, with no pole at zero. In particular, the Taylor series expansion of $F_{\hbar}$ around [math] makes sense, and it gives an element $F\in(\mathscr{U}\mathfrak{g}/\mathscr{U}\mathfrak{g}\cdot\mathfrak{g}_{\lambda})^{\mathbin{\otimes}2}[\!\!\;[\hbar]\!\!\;]$, where the tensor product is the usual (not completed) tensor product. Furthermore, $F$ satisfies (2.28) in $(\mathscr{U}\mathfrak{g}/\mathscr{U}\mathfrak{g}\cdot\mathfrak{g}_{\lambda})^{\mathbin{\otimes}3}[\!\!\;[\hbar]\!\!\;]$ and is $\mathfrak{g}_{\lambda}$-invariant. *

Proof**:**

As mentioned before, $\mathfrak{g}_{\mathrm{i}\lambda/\hbar}$ and \smash{\tilde{\pi}_{\mathrm{i}\lambda/\hbar}^{\pm}\big{|}_{\mathscr{U}({\mathfrak{n}}^{\pm})}}\colon\mathscr{U}({\mathfrak{n}}^{\pm})\to\mathscr{U}({\tilde{\mathfrak{n}}}^{\pm}) are independent of $\hbar$, so only the coefficients $\smash{p_{\mathrm{i}\lambda/\hbar}^{w}(\alpha_{w})^{-1}}$ in the formula for $F_{\mathrm{i}\lambda/\hbar}$ obtained in Subsection 2.2 depend on $\hbar$. Since they are products of elements of the form

[TABLE]

with $\mu\in\mathbbm{N}_{0}\hat{\Delta}^{+}\setminus\{0\}$, their dependence on $\hbar$ is rational without a pole at zero. (Observe that $\frac{1}{2}(\mu,\mu)-(\rho,\mu)$ and $(\mathrm{i}\lambda,\mu)$ cannot vanish simultaneously since $P_{\lambda}$ is assumed to be countable.) Consequently, we may take the Taylor expansion of $F_{\mathrm{i}\lambda/\hbar}$ around $\hbar=0$. To see that this yields an element in the usual tensor product, note that the formal expansion of $\smash{p_{\mathrm{i}\lambda/\hbar}(\mu)^{-1}}$ is a multiple of $\hbar$ unless $(\lambda,\mu)=0$. Now $\smash{p_{\mathrm{i}\lambda/\hbar}^{w}(\alpha_{w})^{-1}}=\smash{\prod_{i=1}^{\lvert w\rvert}p_{\mathrm{i}\lambda/\hbar}(\alpha_{w_{i\dots\lvert w\rvert}})^{-1}}$, and if the formal expansions of both $p_{\mathrm{i}\lambda/\hbar}(\alpha_{w_{i\dots\lvert w\rvert}})^{-1}$ and $p_{\mathrm{i}\lambda/\hbar}(\alpha_{w_{i+1\dots\lvert w\rvert}})^{-1}$ are not multiples of $\hbar$, then $(\lambda,\alpha_{w_{i}})=0$, i.e. $\alpha_{w_{i}}\in\Delta^{\prime}$. However, Lemma 2.8 ensures that this cannot happen too often: If $M$ is the constant obtained in that lemma, then at least $\lceil\lvert w\rvert/(M+1)\rceil$ many elements in the formal expansion of $p_{\mathrm{i}\lambda/\hbar}^{w}(\alpha_{w})^{-1}$ are multiples of $\hbar$, so this expansion is of order at least $\smash{\hbar^{\lceil\lvert w\rvert/(M+1)\rceil}}$. Consequently, only finitely many words contribute to a given order in $\hbar$, so that we do not need to complete the tensor product. Since every $F_{\hbar}$ satisfies (2.28) and is $\mathfrak{g}_{\lambda}$-invariant, this is also true for the formal expansion $F$. $\boxempty$

Let us now apply this theorem to quantize complex coadjoint orbits. Let $G$ be a complex connected semisimple Lie group with coadjoint orbit $\mathcal{O}_{\lambda}$ through a semisimple element $\lambda\in\mathfrak{g}^{*}$. Pick a Cartan subalgebra $\mathfrak{h}$ containing $\lambda^{\sharp}$. Choose an invariant ordering for which $P_{\lambda}$ is countable (e.g. a standard invariant ordering).

By Subsection 1.1 we know that $G_{\lambda}$ is connected. Therefore the $\mathfrak{g}_{\lambda}$-invariance of the elements $F$ and $F_{\hbar}$ constructed previously implies their $G_{\lambda}$-invariance. Consequently we can apply the results of Subsection 1.2 in order to obtain holomorphic $G$-invariant bidifferential operators on $\mathcal{O}_{\lambda}\cong G/G_{\lambda}$. Define the formal product

[TABLE]

which is well-defined since the previous theorem asserts that $F\in(\mathscr{U}\mathfrak{g}/\mathscr{U}\mathfrak{g}\cdot\mathfrak{g}_{\lambda})^{\mathbin{\otimes}2}[\!\!\;[\hbar]\!\!\;]$.

Proposition \theproposition.

The product $\star$ is associative and restricts to a product

[TABLE]

*on formal power series of holomorphic functions. Moreover, $\star$ is $G$-invariant, in the sense that $(g\mathbin{\triangleright}f_{1})\star(g\mathbin{\triangleright}f_{2})=g\mathbin{\triangleright}(f_{1}\star f_{2})$ holds for all $g\in G$ and $f_{1},f_{2}\in\mathscr{C}^{\infty}(\mathcal{O}_{\lambda})[\!\!\;[\hbar]\!\!\;]$. *

Proof**:**

It is a standard argument that the twist condition (2.28) translates into associativity of the induced product. That $\star$ restricts to power series of holomorphic functions and is $G$-invariant is immediate since the image of $\Psi$ consists of holomorphic $G$-invariant bidifferential operators. $\boxempty$

In order to define strict star products from $F_{\hbar}$ directly, i.e. without taking a formal power series expansion, we need to ensure that $\Psi(F_{\hbar})$ is well-defined. To do that we introduce polynomials on the coadjoint orbit. It will turn out that only finitely many elements of the infinite sum defining $F_{\hbar}$ contribute non-trivially when $\Psi(F_{\hbar})$ is applied to polynomials.

Recall from Subsection 1.1 that we may assume without loss of generality that $G$ is a closed complex Lie subgroup of $\mathrm{GL}_{N}(\mathbbm{C})$. We fix a way to realize $G$ as such a matrix Lie group once and for all. In particular, the Lie algebra $\mathfrak{g}$ of $G$ is realized as a complex Lie subalgebra of $\mathfrak{gl}_{N}(\mathbbm{C})$.

Definition \thedefinition (Polynomials on $\mathcal{O}_{\lambda}$).

Let $\mathcal{O}_{\lambda}\subseteq\mathfrak{g}^{*}$ be a complex coadjoint orbit. Then

[TABLE]

*is called the algebra of polynomials on $\mathcal{O}_{\lambda}$. *

Recall that the symmetric algebra $\mathrm{S}\mathfrak{g}$ of $\mathfrak{g}$ is isomorphic (as an algebra) to the algebra $\operatorname{\mathrm{Pol}}(\mathfrak{g}^{*})$ of polynomials on $\mathfrak{g}^{*}$. The isomorphism sends an element $X_{1}\vee\dots\vee X_{j}\in\mathrm{S}^{j}\mathfrak{g}$ to $\xi\mapsto\xi(X_{1})\dots\xi(X_{j})$.

Definition \thedefinition (Polynomials on $G$).

*For a complex linear Lie group $G$, the algebra of polynomials $\operatorname{\mathrm{Pol}}(G)$ is the unital complex subalgebra of $\mathscr{C}^{\infty}(G)$ generated by the functions $P_{ij}\colon G\to\mathbbm{C}$, $g\mapsto g_{ij}$. *

Polynomials on a complex Lie group $G$ are holomorphic. In the case of semisimple connected Lie groups both the Lie group itself and the coadjoint orbit are affine algebraic varieties, see Subsection 1.1, and our definition of polynomials coincides with the definition of regular functions on algebraic varieties. If $G$ is connected and semisimple, then the definition of polynomials on $G$ is independent of the way in which $G$ is realized as a linear group, which can be proven as outlined in Appendix A.2.

Proposition \theproposition.

*Assume that the complex linear Lie group $G$ is semisimple and connected. Then $\pi^{*}\colon\mathrm{Hol}(\mathcal{O}_{\lambda})\cong\mathrm{Hol}(G/G_{\lambda})\to\mathrm{Hol}(G)^{G_{\lambda}}$ restricts to an isomorphism $\pi^{*}\colon\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})\to\operatorname{\mathrm{Pol}}(G)^{G_{\lambda}}$. *

Proof**:**

Since the Lie algebra $\mathfrak{g}$ is semisimple, we have $\mathfrak{g}=[\mathfrak{g},\mathfrak{g}]$, i.e. every element of $\mathfrak{g}$ can be written as a sum of commutators. Consequently the trace of any element of $\mathfrak{g}$ is zero. Therefore any element in a sufficiently small neighbourhood of the identity of $G$ must have determinant $1$, and consequently $G$ is a Lie subgroup of $\mathrm{SL}_{N}(\mathbbm{C})$.

Let $E_{ij}\in\mathfrak{gl}_{N}(\mathbbm{C})$ be the matrix that is $1$ at position $(i,j)$ and [math] otherwise. Extend $\lambda$ to a linear functional $\tilde{\lambda}\in\mathfrak{gl}_{N}(\mathbbm{C})^{*}$. For an element $X\in\mathfrak{g}=\mathrm{S}^{1}\mathfrak{g}$, which we identify with a polynomial on $\mathfrak{g}^{*}$, we compute

[TABLE]

Since $\det g=1$ we can write $(g^{-1})_{ik}$ as a polynomial in the entries of $g$, so that \smash{\pi^{*}\big{(}X|_{\mathcal{O}_{\lambda}}\big{)}} itself is a polynomial in the entries of $g$. Since $\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$ is generated by $\smash{X|_{\mathcal{O}_{\lambda}}}$ and $\pi^{*}$ is an algebra homomorphism, it follows that $\pi^{*}p\in\operatorname{\mathrm{Pol}}(G)$ for any $p\in\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$. Injectivity of $\pi^{*}$ is immediate. Surjectivity is harder to prove. One can either use methods from algebraic geometry (making use of Subsection 1.1, see for example [23, Chapter 12]) or work in a more differential geometric setting using $G$-finite functions as outlined in Appendix A.2. $\boxempty$

Recall the degree $d^{\prime}$ introduced in the proof of Lemma 2.12.

Lemma 2.13**.**

*For any polynomial $p\in\operatorname{\mathrm{Pol}}(\mathrm{GL}_{N}(\mathbbm{C}))$, there is a constant $N_{p}\in\mathbbm{N}$ such that $u^{\mathrm{left},(1,0)}p=v^{\mathrm{left},(1,0)}p=0$ holds for any $u\in\mathscr{U}({\tilde{\mathfrak{n}}}^{+})\subseteq\mathscr{U}(\mathfrak{gl}_{N}(\mathbbm{C}))$ of degree $d^{\prime}$ greater than $N_{p}$ and any $v\in\mathscr{U}({\tilde{\mathfrak{n}}}^{-})\subseteq\mathscr{U}(\mathfrak{gl}_{N}(\mathbbm{C}))$ of degree $d^{\prime}$ smaller than $-N_{p}$. *

Proof**:**

Using the Leibniz rule we may assume that $p=P_{k\ell}$ in the notation of Subsection 2.3. Let $E_{ij}\in\mathfrak{gl}_{N}(\mathbbm{C})$ be the matrix that is $1$ at position $(i,j)$ and [math] otherwise. It is easy to check that $E_{ij}^{\mathrm{left}}P_{k\ell}=\delta_{j\ell}P_{ki}$ and therefore X^{\mathrm{left}}P_{k\ell}=\smash{\big{(}\sum_{i,j}X_{ij}E_{ij}\big{)}^{\mathrm{left}}P_{k\ell}}=\sum_{i}X_{i\ell}P_{ki} for all $X\in\mathfrak{gl}_{N}(\mathbbm{C})$. Since $P_{k\ell}$ is holomorphic, this implies that also $\smash{X^{\mathrm{left},(1,0)}P_{k\ell}}=\smash{X^{\mathrm{left}}P_{k\ell}}=\sum_{i}X_{i\ell}P_{ki}$. Consequently, if $u=u_{1}\dots u_{M}\in\mathscr{U}(\mathfrak{gl}_{N}(\mathbbm{C}))$ with $u_{1},\dots,u_{M}\in\mathfrak{gl}_{N}(\mathbbm{C})$, then

[TABLE]

Since $\operatorname{\mathrm{ad}}_{X}$ is nilpotent for any $X\in{\tilde{\mathfrak{n}}}^{+}$ it follows that $0=(\operatorname{\mathrm{ad}}X)_{s}=\operatorname{\mathrm{ad}}(X_{s})$ for $X\in{\tilde{\mathfrak{n}}}^{+}$, where the index $s$ stands for the semisimple part of the Jordan decomposition. Since $\mathfrak{g}$ is semisimple this implies $X_{s}=0$, so every $X\in{\tilde{\mathfrak{n}}}^{+}$ is realized by a nilpotent matrix. It follows from Engel’s theorem that any matrix Lie algebra consisting of nilpotent matrices is nilpotent as an algebra, so there exists a constant $M\in\mathbbm{N}$ such that products of $M$ or more elements of ${\tilde{\mathfrak{n}}}^{+}$ vanish. Therefore, if $u$ is a product of at least $M$ elements of ${\tilde{\mathfrak{n}}}^{+}$ the above calculation shows that $u^{\mathrm{left}}P_{k\ell}=0$. If $M^{\prime}$ is an upper bound for the degree $d^{\prime}$ of elements of ${\tilde{\mathfrak{n}}}^{+}$ then we can set $N_{P_{k\ell}}\coloneqq MM^{\prime}$. It is easy to check that this constant also works for ${\tilde{\mathfrak{n}}}^{-}$. $\boxempty$

Corollary \thecorollary.

*For all $p,q\in\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$ and all $\hbar\in\mathbbm{C}\setminus P_{\lambda}$, the sum $\sum_{\ell=0}^{\infty}\Psi(F_{\hbar,\ell})(p,q)$ is finite, and $\sum_{\ell=0}^{\infty}\Psi(F_{\hbar,\ell})(p,q)\in\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$. *

Proof**:**

Subsection 2.3* implies that $\pi^{*}p$ and $\pi^{*}q$ are polynomials. By Lemma 2.12 the components $F_{\hbar,\ell}$ are of degree $(\ell,-\ell)$, and then the previous lemma implies that only finitely many summands of $\sum_{\ell=0}^{\infty}F^{\mathrm{left},(1,0)}_{\hbar,\ell}(\pi^{*}p,\pi^{*}q)$ are non-zero. Its proof shows that $\smash{\sum_{\ell=0}^{\infty}F^{\mathrm{left},(1,0)}_{\hbar,\ell}(\pi^{*}p,\pi^{*}q)}$ is again a polynomial. The components $F_{\hbar,\ell}$ are $\mathfrak{g}_{\lambda}$-invariant, and therefore, since $G_{\lambda}$ is connected by Subsection 1.1, also $G_{\lambda}$-invariant. Consequently $\sum_{\ell=0}^{\infty}F^{\mathrm{left},(1,0)}_{\hbar,\ell}(\pi^{*}p,\pi^{*}q)$ is $G_{\lambda}$-invariant by Lemma 1.1. Finally, $\sum_{\ell=0}^{\infty}\Psi(F_{\hbar,\ell})(p,q)=\sum_{\ell=0}^{\infty}\pi_{*}({F^{\mathrm{left},(1,0)}_{\hbar,\ell}}^{\,\!}(\pi^{*}p,\pi^{*}q))$ is a polynomial by Subsection 2.3. $\boxempty$*

Corollary \thecorollary.

Let $\mathcal{O}_{\lambda}$ be a semisimple coadjoint orbit of a complex connected semisimple Lie group $G$ with Lie algebra $\mathfrak{g}$. Assume that $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$ containing $\lambda^{\sharp}$, and that one has chosen an invariant ordering. Then for any $\hbar\in\mathbbm{C}\setminus P_{\lambda}$,

[TABLE]

*defines an associative and $G$-invariant product (where $G$-invariant means that $(g\mathbin{\triangleright}p)*_{\hbar}(g\mathbin{\triangleright}q)=g\mathbin{\triangleright}(p*_{\hbar}q)$ holds for any $g\in G$ and $p,q\in\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$). For $p,q\in\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$, $p*_{\hbar}q$ depends rationally on $\hbar$, and the formal expansion of $*_{\hbar}$ around $\hbar=0$ coincides with the formal product $\star$. *

Proof**:**

As in the formal case, it is a standard argument to show that (2.28) implies the associativity of $*_{\hbar}$. Since the codomain of $\Psi$ consists of $G$-invariant bidifferential operators, it is clear that $*_{\hbar}$ is $G$-invariant. Since the dependence of $F_{\hbar}$ on $\hbar$ is rational without pole at [math], it follows that $*_{\hbar}$ also depends rationally on $\hbar$ without pole at [math], and since $\star$ was constructed from the formal expansion of $F_{\hbar}$, it coincides with the formal expansion of $*_{\hbar}$. $\boxempty$

Remark \theremark.

When considering $\Psi(F_{\hbar,\ell})$, we may leave out the projections $\tilde{\pi}_{\lambda}^{\pm}$ in the formula for $F_{\hbar,\ell}$ from Lemma 2.12 to obtain the same result. Indeed, by Lemma 2.10 the difference of $F_{\hbar,\ell}$ and

[TABLE]

is an element in the ideal $\mathscr{U}\mathfrak{g}\cdot\mathfrak{g}_{\lambda}\mathbin{\otimes}\mathscr{U}\mathfrak{g}+\mathscr{U}\mathfrak{g}\mathbin{\otimes}\mathscr{U}\mathfrak{g}\cdot\mathfrak{g}_{\lambda}$ and therefore contained in the kernel of $\Psi$ by Lemma 1.1.

Recall that we obtained a condition for $P_{\lambda}$ being countable in Subsection 2.3, and that this condition is satisfied in particular when the ordering is standard, see Subsection 2.3.

Proposition \theproposition.

*Assume that $P_{\lambda}$ is countable. Then the first order commutator of $\star$ coincides with the Poisson bracket induced by the KKS form $\omega_{\mathrm{KKS}}$ defined in (1.2). *

Proof**:**

Note that the formal expansion of

[TABLE]

is of order $\hbar$ if $(\lambda,\mu)\neq 0$. It follows from Subsection 2.2 that the element $F$ is the formal expansion of

[TABLE]

Using that the words $w\in\tilde{W}$ with $\lvert w\rvert\leq 1$ are precisely the empty word and the one-letter words $(\ell)$ with $\alpha_{\ell}\in\hat{\Delta}^{+}$, i.e. $(\lambda,\alpha_{\ell})\neq 0$, the first sum expands to $1+\mathrm{i}\hbar\sum_{\alpha\in\hat{\Delta}^{+}}(\lambda,\alpha)^{-1}X_{\alpha}\mathbin{\otimes}Y_{\alpha}+\mathcal{O}(\hbar^{2})$. Let us argue why the formal expansion of the second sum is of order $\hbar^{2}$. By definition $\smash{p_{\mathrm{i}\lambda/\hbar}^{w}(\alpha_{w})^{-1}}=\smash{\prod_{i=1}^{\lvert w\rvert}p_{\mathrm{i}\lambda/\hbar}(\alpha_{w_{i\dots\lvert w\rvert}})^{-1}}$. Since, by definition of $\smash{\tilde{W}}$, we have $\smash{\alpha_{w_{\lvert w\rvert}}}\in\hat{\Delta}^{+}$, it is clear that the formal expansions of all summands with $(\lambda,\smash{\alpha_{w_{\lvert w\rvert-1}}}+\smash{\alpha_{w_{\lvert w\rvert}}})\neq 0$ are of order $\hbar^{2}$ (because both $p_{\mathrm{i}\lambda/\hbar}(\smash{\alpha_{(w_{\lvert w\rvert-1},w_{\lvert w\rvert})}})^{-1}$ and $p_{\mathrm{i}\lambda/\hbar}(\smash{\alpha_{w_{\lvert w\rvert}}})^{-1}$ are of order $\hbar$). So assume $(\lambda,\alpha_{w_{\lvert w\rvert-1}}+\alpha_{w_{\lvert w\rvert}})=0$, in which case $\alpha_{w_{\lvert w\rvert-1}}\in\smash{\hat{\Delta}^{+}}$ and, by invariance of the ordering, $\alpha_{w_{\lvert w\rvert-1}}+\alpha_{w_{\lvert w\rvert}}$ is not a root. Therefore $X_{w_{\lvert w\rvert-1}}X_{w_{\lvert w\rvert}}=X_{w_{\lvert w\rvert}}X_{w_{\lvert w\rvert-1}}$, and if $w^{\prime}=(w_{1},\dots,w_{\lvert w\rvert-2},w_{\lvert w\rvert},w_{\lvert w\rvert-1})$ is the word obtained form $w$ by switching the last two letters then $X_{w}=X_{w^{\prime}}$. Similarly $Y_{w}=Y_{w^{\prime}}$. Furthermore, by definition of $\alpha_{w}$, we have $\smash{\alpha_{w_{i\dots\lvert w\rvert}}}=\smash{\alpha_{w^{\prime}_{i\dots\lvert w^{\prime}\rvert}}}$ for all $i<\lvert w\rvert$ and

[TABLE]

But under our assumptions $(\alpha_{w_{\lvert w\rvert}},\lambda)^{-1}+(\alpha_{w_{\lvert w\rvert-1}},\lambda)^{-1}=0$, and therefore the formal expansion of $p_{\mathrm{i}\lambda/\hbar}(\alpha_{w_{\lvert w\rvert}})^{-1}+p_{\mathrm{i}\lambda/\hbar}(\alpha_{w_{\lvert w\rvert-1}})^{-1}$ is $\mathrm{i}\hbar(\alpha_{w_{\lvert w\rvert}},\lambda)^{-1}+\mathrm{i}\hbar(\alpha_{w_{\lvert w\rvert-1}},\lambda)^{-1}+\mathcal{O}(\hbar^{2})=\mathcal{O}(\hbar^{2})$. Consequently, the summands which could potentially be of order $\hbar$ in the sum over $w\in\tilde{W}$ with $\lvert w\rvert\geq 2$ cancel out, and this sum is therefore of order $\hbar^{2}$ as claimed.

To conclude the proof, note that antisymmetrizing the first order gives indeed

[TABLE]

where $\pi_{\mathrm{KKS}}$ denotes the Poisson tensor associated to the KKS symplectic form. $\boxempty$

We conclude this subsection by saying a bit more about the directions in which $\star$ and $*_{\hbar}$ differentiate.

Lemma 2.14**.**

For any $\xi=\operatorname{\mathrm{Ad}}_{g}^{*}\lambda\in\mathcal{O}_{\lambda}$, the subspaces

[TABLE]

*are independent of the choice of $g\in G$. *

Proof**:**

Any two choices $g,g^{\prime}\in G$ differ by an element of $G_{\lambda}$, that is $g^{\prime}=gx$ with $x\in G_{\lambda}$. So it suffices to prove that $\operatorname{\mathrm{span}}\{\operatorname{\mathrm{Ad}}_{x}X_{\alpha}\,,\,\,\alpha\in\hat{\Delta}^{\pm}\}=\operatorname{\mathrm{span}}\{X_{\alpha}\,,\,\,\alpha\in\hat{\Delta}^{\pm}\}$. This follows from the invariance of the ordering and the connectedness of $G_{\lambda}$. $\boxempty$

Therefore the distributions $L_{+}$ and $L_{-}$ in $\mathrm{T}\mathcal{O}_{\lambda}$ spanned by $L_{+,\xi}$ and $L_{-,\xi}$, respectively, are well-defined.

Corollary \thecorollary.

*The star product $*_{\hbar}$ derives the first argument only in the directions of $L_{+}^{(1,0)}$ and the second argument only in the directions of $\smash{L_{-}^{(1,0)}}$. *

Proof**:**

This follows from the formula for $F_{\hbar}$ obtained in Subsection 2.2, from Subsection 2.3, and from Subsection 1.2. $\boxempty$

2.4 Examples

In this subsection we derive formulas for $F_{\hbar}$ in the case $G=\mathrm{SL}_{1+n}(\mathbbm{C})$ for the largest non-trivial stabilizer $G_{\lambda}$. When restricting to real coadjoint orbits in Subsection 4.4, this example allows us to obtain quantizations of complex projective spaces and hyperbolic discs.

Example \theexample ($\mathrm{SL}_{1+n}(\mathbbm{C})$).

Let $G=\mathrm{SL}_{1+n}(\mathbbm{C})$ be the Lie group of matrices with determinant $1$. Its Lie algebra $\mathfrak{g}=\mathfrak{sl}_{1+n}(\mathbbm{C})$ consists of matrices with trace [math]. Number the rows and columns of a matrix $X\in\mathfrak{g}$ by $0,\dots,n$. Let $\lambda\colon\mathfrak{g}\to\mathbbm{C}$, $X\mapsto-\mathrm{i}rX_{0,0}$ where $r\in\mathbbm{C}$. Using that the Killing form $B$ satisfies $B(X,Y)=2(n+1)\operatorname{tr}(XY)$, where $\operatorname{tr}$ is the usual (not normalized) matrix trace, it follows that $\lambda^{\sharp}$ is a multiple of the diagonal matrix $\operatorname{\mathrm{diag}}(n,-1,\dots,-1)$, and therefore

[TABLE]

We choose the Cartan subalgebra $\mathfrak{h}$ consisting of the diagonal matrices in $\mathfrak{g}$. The roots are then given by $\alpha_{i,j}=L_{i}-L_{j}$ for $0\leq i,j\leq n$ with $i\neq j$, where $L_{i}\in\mathfrak{h}^{*}$, $L_{i}(X)=X_{i,i}$. If we let the roots $\alpha_{i,j}$ with $i<j$ be positive, then the simple roots are $\alpha_{0,1},\alpha_{1,2},\dots,\alpha_{n-1,n}$. As before, denote the matrix with entry $1$ at position $(i,j)$ by $E_{i,j}$, and define $X_{i,j}\coloneqq E_{i,j}\in\mathfrak{g}^{\alpha_{i,j}}$ and $Y_{i,j}\coloneqq E_{j,i}\in\mathfrak{g}^{\alpha_{j,i}}=\mathfrak{g}^{-\alpha_{i,j}}$. Note that $B(X_{i,j},Y_{i,j})=2(n+1)\operatorname{tr}(X_{i,j}Y_{i,j})=2(n+1)$, so we use a normalization different from that in Subsection 2.1.

If $n=1$, it is easy to simplify the formula for $F_{\hbar}$ obtained in Subsection 2.2: There is only one positive root $\alpha=\alpha_{0,1}$, and there is a unique word $w_{\ell}$ of a given length $\ell\in\mathbbm{N}_{0}$. Note that $\lambda=-\mathrm{i}r\alpha/2$ and $\rho=\alpha/2$, so $p_{\mathrm{i}\lambda/\hbar}(m\alpha)=\frac{1}{2}m^{2}(\alpha,\alpha)-\frac{1}{2}m(\alpha,\alpha)-\frac{1}{2\hbar}mr(\alpha,\alpha)=\frac{1}{4}m(m-1-\frac{r}{\hbar})$. Therefore

[TABLE]

We set $X\coloneqq X_{0,1}$ and $Y\coloneqq Y_{0,1}$. Since $B(X,Y)=4$ we have to plug the normalized elements $X/2$ and $Y/2$ into (2.27), and obtain

[TABLE]

This result was already obtained in [1, Example 4.16], but the following result for arbitrary $n$ is new. We prove it by computing the canonical element of the Shapovalov pairing directly, instead of simplifying (2.27).

Proposition \theproposition.

For $G=\mathrm{SL}_{1+n}(\mathbbm{C})$, the same $\lambda$ and the same ordering as above, one has

[TABLE]

Proof**:**

The Lie algebras ${\tilde{\mathfrak{n}}}^{+}$ and ${\tilde{\mathfrak{n}}}^{-}$ are commutative Lie algebras spanned by $X_{0,1},\dots,X_{0,n}$ and $Y_{0,1},\dots,Y_{0,n}$ respectively, and therefore $\{X^{I}\coloneqq X_{0,1}^{I_{1}}\dots X_{0,n}^{I_{n}}\mid I\in\mathbbm{N}_{0}^{n}\}$ and $\{Y^{J}\coloneqq Y_{0,1}^{J_{1}}\dots Y_{0,n}^{J_{n}}\mid J\in\mathbbm{N}_{0}^{n}\}$ are bases of $\mathscr{U}({\tilde{\mathfrak{n}}}^{+})$ and $\mathscr{U}({\tilde{\mathfrak{n}}}^{-})$. The Lie algebra ${\mathfrak{n}}^{+}$ is spanned by $X_{i,j}$ with $i<j$ and we can view $X^{I}$ also as an element of $\mathscr{U}({\mathfrak{n}}^{+})$. Then $\smash{\tilde{\pi}_{\lambda}^{+}(X^{I})}=X^{I}$ and similarly $\smash{\tilde{\pi}_{\lambda}^{-}(Y^{J})}=Y^{J}$. Consequently $\smash{\langle X^{I},Y^{J}\rangle^{\sim}_{\mathrm{i}\lambda/\hbar}}=\smash{\langle X^{I},Y^{J}\rangle_{\mathrm{i}\lambda/\hbar}}$. For degree reasons the bases $\{X^{I}\}$ and $\{Y^{J}\}$ are orthogonal, meaning that $\smash{\langle X^{I},Y^{J}\rangle_{\mathrm{i}\lambda/\hbar}}=0$ for $I\neq J$. Indeed, $X^{I}$ and $Y^{J}$ are homogeneous with respect to the degree $d$ defined in the beginning of Subsection 2.1, $d(X^{I})=I_{1}d(X_{0,1})+\dots+I_{n}d(X_{0,n})=I_{1}\alpha_{0,1}+\dots+I_{n}\alpha_{0,n}$, and $d(Y^{J})=-(J_{1}\alpha_{0,1}+\dots+J_{n}\alpha_{0,n})$. Since the $\alpha_{0,i}$ are linearly independent, Lemma 2.2 implies the claimed orthogonality.

Therefore it suffices to determine the normalization $\langle X^{I},Y^{I}\rangle_{\mathrm{i}\lambda/\hbar}$. Define $H_{i}\coloneqq[X_{0,i},Y_{0,i}]=E_{0,0}-E_{i,i}$. Given a multiindex $I\in\mathbbm{N}_{0}^{n}$ we can form a sequence that starts with $I_{1}$ many $1$’s, then has $I_{2}$ many $2$’s, $\dots$, then $I_{n}$ many $n$’s. Denote the $k$-th element of this sequence by $I_{\{k\}}$. Introduce the projection $({}\cdot{})_{0}$ to $\mathscr{U}\mathfrak{h}$ in the decomposition $\mathscr{U}\mathfrak{g}=\mathscr{U}\mathfrak{h}\oplus({\mathfrak{n}}^{-}\cdot\mathscr{U}\mathfrak{g}+\mathscr{U}\mathfrak{g}\cdot{\mathfrak{n}}^{+})$, so that $\pi_{\lambda}(u)=\lambda((u)_{0})$. Then we claim that

[TABLE]

To see that this formula implies the proposition, note that

[TABLE]

and that $\frac{\mathrm{i}}{\hbar}\lambda(H_{i})=\frac{r}{\hbar}$ for all $i=1,\dots,n$. So

[TABLE]

and an application of the multinomial theorem gives (2.39).

It remains to prove (2.40). For $n=1$ this is the statement of [16, Lemma 5.2]. Note that this also means that $Z\coloneqq X_{0,n}^{I_{n}}Y_{0,n}^{I_{n}}-I_{n}!H_{n}(H_{n}-1)\dots(H_{n}-I_{n}+1)\in\mathscr{U}(\operatorname{\mathrm{span}}\{X_{0,n},Y_{0,n},H_{n}\})$ satisfies $(Z)_{0}=0$. We proceed by induction and assume that (2.40) holds for $n-1$. Writing $I_{-}=(I_{1},\dots,I_{n-1},0)$ and noting that $[H_{n},X_{0,i}]=X_{0,i}$ for $1\leq i\leq n-1$, we compute

[TABLE]

Since $(Z)_{0}=0$ and $d(Z)=d(X_{0,n}^{I_{n}}Y_{0,n}^{I_{n}}-I_{n}!H_{n}(H_{n}-1)\dots(H_{n}-I_{n}+1))=0$ we can write $Z=Y_{0,n}Z^{\prime}X_{0,n}$ for some $Z^{\prime}\in\mathscr{U}(\operatorname{\mathrm{span}}\{X_{0,n},Y_{0,n},H_{n}\})$. Since $Y_{0,n}\in\mathfrak{g}^{\alpha_{n,0}}$ any commutator of $Y_{0,n}$ with elements of $\mathfrak{g}^{\alpha_{0,1}},\dots,\mathfrak{g}^{\alpha_{0,n-1}}$ has degree $d$ equal to $L_{n}-\sum_{i=0}^{n-1}c_{i}L_{i}$ for some $c_{i}\in\mathbbm{Z}$, so it must either be [math] or in a negative root space. Therefore $(X^{I_{-}}ZY^{I-})_{0}=0$, and the claim follows by applying the induction hypothesis to the first summand in the equation above. $\boxempty$

Corollary \thecorollary.

Let $G=\mathrm{SL}_{1+n}(\mathbbm{C})$ and $\lambda$ be as above, but choose the opposite ordering, for which $\alpha_{i,j}$ with $i>j$ is positive. Then

[TABLE]

Proof**:**

The only change in the computation above is that the roles of $X_{0,i}$ and $Y_{0,i}$ are swapped. Now $[Y_{0,i},X_{0,i}]=E_{i,i}-E_{0,0}$, so $\frac{\mathrm{i}}{\hbar}\lambda([Y_{0,i},X_{0,i}])=-\frac{r}{\hbar}$, which means that $r$ changes sign. $\boxempty$

3 Continuity

In this section, we will extend the product $*_{\hbar}\colon\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})\times\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})\to\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$ obtained in Subsection 2.3 to a product $*_{\hbar}\colon\mathrm{Hol}(\mathcal{O}_{\lambda})\times\mathrm{Hol}(\mathcal{O}_{\lambda})\to\mathrm{Hol}(\mathcal{O}_{\lambda})$ on all holomorphic functions on the coadjoint orbit, that is continuous with respect to the topology of locally uniform convergence. More precisely, we prove the following theorem.

Theorem \thethm.

*Let $\mathcal{O}_{\lambda}$ be a complex semisimple coadjoint orbit of a complex semisimple connected Lie group $G$. Then for any $\hbar\in\mathbbm{C}\setminus P_{\lambda}$ the product $*_{\hbar}$ on $\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$ is continuous with respect to the topology of locally uniform convergence and extends to a continuous and $G$-invariant product $*_{\hbar}\colon\mathrm{Hol}(\mathcal{O}_{\lambda})\times\mathrm{Hol}(\mathcal{O}_{\lambda})\to\mathrm{Hol}(\mathcal{O}_{\lambda})$ on the space of all holomorphic functions on $\mathcal{O}_{\lambda}$. *

The proof of this theorem proceeds as follows: In Subsection 3.1 we prove the continuity of $*_{\hbar}$ with respect to a topology that we call the reduction-topology and in Subsection 3.3 we prove that the reduction-topology coincides with the topology of locally uniform convergence. Consequently $*_{\hbar}$ extends to the completion of the space of polynomials on $\mathcal{O}_{\lambda}$. Using the results of Subsection 3.2 we prove in Subsection 3.3 that this completion is the space $\mathrm{Hol}(\mathcal{O}_{\lambda})$ of all holomorphic functions on $\mathcal{O}_{\lambda}$.

In the whole section we assume that the complex connected semisimple Lie group $G$ is concretely realized as a complex Lie subgroup of $\mathrm{GL}_{N}(\mathbbm{C})$ for some $N\in\mathbbm{N}$, as explained in Subsection 1.1. In particular, since $G$ is semisimple, it is a closed submanifold of $\mathbbm{C}^{N\times N}$.

3.1 Continuity in the reduction-topology

In this subsection we prove the continuity of the star product $*_{\hbar}$ with respect to a topology that we call the reduction-topology, and which is defined below. Recall that a sequence of functions $f_{i}\colon X\to\mathbbm{C}$ on a topological space $X$ is said to be locally uniformly convergent if for every $x\in X$ there is a neighbourhood $U\subseteq X$ such that $f_{i}$ converges uniformly to $f$ on $U$, i.e. $\lim_{i\to\infty}\sup_{y\in U}\lvert f_{i}(y)-f(y)\rvert=0$. In this work, $X$ will always be a manifold. Then the topology of locally uniform convergence coincides with the topology of compact convergence (for every compact subset $K\subseteq X$, $f_{i}$ converges uniformly on $K$), and is therefore a locally convex topology, defined by the seminorms $\lVert f\rVert_{K}\coloneqq\sup_{K}\lvert f\rvert$.

Denote the ideal of polynomials in $\operatorname{\mathrm{Pol}}(\mathbbm{C}^{N\times N})$ whose restriction to $G$ vanishes by $\mathcal{I}(G)$.

Definition \thedefinition (Reduction-topology).

*The topology $\mathcal{T}_{\mathrm{lc}}$ of locally uniform convergence on the space $\operatorname{\mathrm{Pol}}(\mathbbm{C}^{N\times N})$ of polynomials on $\mathbbm{C}^{N\times N}$ induces a quotient topology on the space $\operatorname{\mathrm{Pol}}(G)\cong\operatorname{\mathrm{Pol}}(\mathbbm{C}^{N\times N})/\mathcal{I}(G)$ of polynomials on $G$, and we call the subspace topology on the space $\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})\cong\operatorname{\mathrm{Pol}}(G)^{G_{\lambda}}$ of polynomials on the coadjoint orbit $\mathcal{O}_{\lambda}$ the reduction-topology. *

In Subsection 3.3 we will prove that the reduction-topology coincides with the topology of locally uniform convergence on $\mathcal{O}_{\lambda}$.

This topology is convenient for obtaining continuity estimates for $\mathbin{*_{\hbar}}$, since we gave a description of $\Psi(F_{\hbar})$ via bidifferential operators on $G$ in Subsection 1.2. Since we assume that the Lie group $G$ is concretely realized as a complex Lie subgroup of $\mathrm{GL}_{N}(\mathbbm{C})$, its Lie algebra $\mathfrak{g}$ is realized as a Lie subalgebra of $\mathfrak{gl}_{N}(\mathbbm{C})$. Considering the element $F^{\prime}_{\hbar,\ell}$ defined in (2.35) as an element of $\mathscr{U}(\mathfrak{gl}_{N}(\mathbbm{C}))\mathbin{\otimes}\mathscr{U}(\mathfrak{gl}_{N}(\mathbbm{C}))$, we let

[TABLE]

This is well-defined since the sum over $\ell$ is finite by Lemma 2.13 and $(F^{\prime}_{\hbar,\ell})^{\mathrm{left},(1,0)}(p,q)$ is again a polynomial. Note that $*^{\prime}_{\hbar}$ is (in general) not associative since $\sum_{\ell=0}^{\infty}\smash{F^{\prime}_{\hbar,\ell}}$ satisfies (2.28) only after passing to the quotient. However, since $\smash{F^{\prime}_{\hbar,\ell}}$ lies in the subspace $\mathscr{U}\mathfrak{g}\mathbin{\otimes}\mathscr{U}\mathfrak{g}$ it induces a product on $\operatorname{\mathrm{Pol}}(G)\cong\operatorname{\mathrm{Pol}}(\mathbbm{C}^{N\times N})/\mathcal{I}(G)$. As in Subsection 2.3 it follows that the restriction of this product to $\operatorname{\mathrm{Pol}}(G)^{G_{\lambda}}\cong\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$ coincides with $\mathbin{*_{\hbar}}$.

Theorem \thethm.

*For $\hbar\in\mathbbm{C}\setminus P_{\lambda}$ the product $*^{\prime}_{\hbar}$ on $\operatorname{\mathrm{Pol}}(\mathbbm{C}^{N\times N})$ is continuous with respect to the topology of locally uniform convergence $\mathcal{T}_{\mathrm{lc}}$. *

Before proving this theorem in the rest of this section, we would like to note the following consequence, which motivates the definition of the reduction-topology given above.

Corollary \thecorollary.

*For $\hbar\in\mathbbm{C}\setminus P_{\lambda}$ the product $\mathbin{*_{\hbar}}$ on $\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$ is continuous with respect to the reduction-topology. *

Proof**:**

This follows immediately from the previous theorem and the construction of the reduction-topology. $\boxempty$

Remark \theremark.

It is interesting to point out that the proof of Subsection 3.1 will not use anything about the actual Lie algebra structure but semisimplicity and the form of the element $F_{\hbar}$. In fact, we only need that the coefficients of $F_{\hbar}$ behave like $p^{w}_{\lambda}(\alpha_{w})\approx\lvert w\rvert^{2}$ for large $\lvert w\rvert$. The rest of the proof consists in counting terms and checking that there are not too many.

The strategy to prove Subsection 3.1 is as follows. We first introduce a different locally convex topology that is better suited for obtaining continuity estimates. Then we prove that this topology is equivalent to the topology of locally uniform convergence and we prove the continuity of $*^{\prime}_{\hbar}$ with respect to this topology.

Set $m=N^{2}$. Let $B=\{b_{1},\dots,b_{m}\}$ be the standard basis of $\mathbbm{C}^{m}$ and denote the dual basis of $(\mathbbm{C}^{m})^{*}$ by $B^{*}=\{b_{1}^{*},\dots,b_{m}^{*}\}$. Elements of $\operatorname{\mathrm{Pol}}(\mathbbm{C}^{m})\cong\mathrm{S}((\mathbbm{C}^{m})^{*})$ (where $\mathrm{S}$ denotes the symmetric tensor algebra) can be written uniquely in the form $\sum_{I\in\mathbbm{N}_{0}^{m}}a_{I}b_{I}^{*}$. Here $I\in\mathbbm{N}_{0}^{m}$ is a multiindex, $\smash{b_{I}^{*}}=\smash{(b_{1}^{*})^{\vee I_{1}}}\vee\dots\vee\smash{(b_{m}^{*})^{\vee I_{m}}}$ and only finitely many of the coefficients $a_{I}\in\mathbbm{C}$ are non-zero. For any $R\in\mathbbm{R}^{+}$ define a norm

[TABLE]

Note that these norms coincide with the $T_{0}$-norms with respect to the basis $B^{*}$, studied for example in [40]. We denote the locally convex topology given by endowing $\operatorname{\mathrm{Pol}}(\mathbbm{C}^{m})\cong\mathrm{S}((\mathbbm{C}^{m})^{*})$ with the seminorms {\mathopen{}\mathclose{{}\left|\kern-1.07639pt\mathopen{}\mathclose{{}\left|\kern-1.07639pt\mathopen{}\mathclose{{}\left|{}\cdot{}}\right|\kern-1.07639pt}\right|\kern-1.07639pt}\right|}_{R} by \mathcal{T}_{{\mathopen{}\mathclose{{}\left|\kern-0.75346pt\mathopen{}\mathclose{{}\left|\kern-0.75346pt\mathopen{}\mathclose{{}\left|{}\cdot{}}\right|\kern-0.75346pt}\right|\kern-0.75346pt}\right|}}. This topology can equivalently be defined by the countable set of norms {\mathopen{}\mathclose{{}\left|\kern-1.07639pt\mathopen{}\mathclose{{}\left|\kern-1.07639pt\mathopen{}\mathclose{{}\left|{}\cdot{}}\right|\kern-1.07639pt}\right|\kern-1.07639pt}\right|}_{R} with $R\in\mathbbm{N}$.

Note that {\mathopen{}\mathclose{{}\left|\kern-1.07639pt\mathopen{}\mathclose{{}\left|\kern-1.07639pt\mathopen{}\mathclose{{}\left|{}\cdot{}}\right|\kern-1.07639pt}\right|\kern-1.07639pt}\right|}_{R} is submultiplicative with respect to the classical product:

[TABLE]

Proposition \theproposition.

*The topologies \mathcal{T}_{{\mathopen{}\mathclose{{}\left|\kern-0.75346pt\mathopen{}\mathclose{{}\left|\kern-0.75346pt\mathopen{}\mathclose{{}\left|{}\cdot{}}\right|\kern-0.75346pt}\right|\kern-0.75346pt}\right|}} and $\mathcal{T}_{\mathrm{lc}}$ coincide. *

Proof**:**

Assume $p=\sum_{I\in\mathbbm{N}_{0}^{m}}a_{I}b_{I}^{*}\in\operatorname{\mathrm{Pol}}(\mathbbm{C}^{m})$ is a polynomial. Given a compact subset $K\subseteq\mathbbm{C}^{m}$, choose $R\in\mathbbm{R}$ such that $\lvert z\rvert\leq R$ holds for all $z\in K$. Then on the one hand we have

[TABLE]

On the other hand, if D_{R}=\{(z_{1},\dots,z_{m})\in\mathbbm{C}^{m}\mid\lvert z_{i}\rvert\leq R\text{ for all i=1,\dots,m}\}\subseteq\mathbbm{C}^{m} denotes a closed polydisc of radius $R$, then Cauchy’s integral formula yields

[TABLE]

Applying this estimate for a polydisc of radius $2mR$ yields

[TABLE]

Consequently we can estimate any norm of \mathcal{T}_{{\mathopen{}\mathclose{{}\left|\kern-0.75346pt\mathopen{}\mathclose{{}\left|\kern-0.75346pt\mathopen{}\mathclose{{}\left|{}\cdot{}}\right|\kern-0.75346pt}\right|\kern-0.75346pt}\right|}} by a seminorm of $\mathcal{T}_{\mathrm{lc}}$ and vice versa, so the topologies \mathcal{T}_{{\mathopen{}\mathclose{{}\left|\kern-0.75346pt\mathopen{}\mathclose{{}\left|\kern-0.75346pt\mathopen{}\mathclose{{}\left|{}\cdot{}}\right|\kern-0.75346pt}\right|\kern-0.75346pt}\right|}} and $\mathcal{T}_{\mathrm{lc}}$ coincide. $\boxempty$

Because of the previous proposition we can and will work with the norms {\mathopen{}\mathclose{{}\left|\kern-1.07639pt\mathopen{}\mathclose{{}\left|\kern-1.07639pt\mathopen{}\mathclose{{}\left|{}\cdot{}}\right|\kern-1.07639pt}\right|\kern-1.07639pt}\right|}_{R} instead of the seminorms $\lVert{}\cdot{}\rVert_{K}$ in the following. To obtain continuity estimates, we need to estimate the coefficients $p_{\lambda}(\mu)$ defined in (2.9).

Lemma 3.1** (Estimates for $p_{\lambda}$).**

For any fixed compact set $K\subseteq\mathfrak{h}^{*}$ there are constants $C>0$ and $M$ such that $p_{\lambda}(\alpha_{w})$ defined in (2.9) satisfies

[TABLE]

*for all words $w\in W$ of length $\lvert w\rvert\geq M$ and all $\lambda\in K$. *

Proof**:**

Assume that the positive roots $\alpha_{1},\dots,\alpha_{k}\in\Delta^{+}$ are ordered in such a way that $\alpha_{1},\dots,\alpha_{r}$ are the simple roots. Write $\alpha_{w}=\sum_{i=1}^{r}c_{w,i}\alpha_{i}$ as a linear combination of simple roots, where $c_{w,i}\in\mathbbm{N}_{0}$ satisfy $\lvert w\rvert\leq\sum_{i=1}^{r}c_{w,i}\leq c\lvert w\rvert$ with $c$ depending only on the root system. Since $(\rho,\alpha_{i})>0$ for all $1\leq i\leq r$ we can choose $c_{\rho},C_{\rho}\in\mathbbm{R}^{+}$ such that $c_{\rho}\leq(\rho,\alpha_{i})\leq C_{\rho}$ holds for all $1\leq i\leq r$. Similarly, there is $C^{\prime}\in\mathbbm{R}^{+}$ with $\lvert(\lambda,\alpha_{i})\rvert\leq C^{\prime}$ for all $\lambda\in K$ and $1\leq i\leq r$. Then

[TABLE]

and for all $\lambda\in K$ we obtain

[TABLE]

Setting $C\coloneqq\frac{1}{4(\rho,\rho)}c_{\rho}^{2}$, $C_{1}\coloneqq c(C_{\rho}+C^{\prime})$, and $M\coloneqq\frac{C_{1}}{C}$, and assuming $\lvert w\rvert\geq M$ we obtain

[TABLE]

$\boxempty$

Corollary \thecorollary (Estimates for $p_{\lambda}^{w}$).

Fix $\lambda\in\mathfrak{h}^{*}$. For any compact set $K\subseteq\mathbbm{C}\setminus P_{\lambda}$ there is a constant $C_{p}>0$ such that $p^{w}_{\mathrm{i}\lambda/\hbar}(\alpha_{w})$ defined in (2.9) satisfies

[TABLE]

*for all words $w\in\tilde{W}$ and all $\hbar\in K$. *

Proof**:**

Note that $K^{\prime}=\{\mathrm{i}\lambda/\hbar\mid\hbar\in K\}$ is a compact subset of $\tilde{\Lambda}$. Let $M$ and $C$ be the constants obtained by applying the previous lemma to $K^{\prime}$, so $\lvert p_{\lambda^{\prime}}(\alpha_{w})\rvert\geq C\lvert w\rvert^{2}$ for all $w\in W$ with $\lvert w\rvert\geq M$ and all $\lambda^{\prime}\in K^{\prime}$. Since $\mathrm{i}\lambda/\hbar\in\tilde{\Lambda}$, we have $\min_{w\in\tilde{W},\lvert w\rvert<M}\lvert p_{\mathrm{i}\lambda/\hbar}(\alpha_{w})\rvert>0$ for all $\hbar\in K$. Since this quantity depends continuously on $\hbar$ the minimum for $\hbar\in K$ exists and must also be positive. Hence we may decrease the constant $C$ such that $\lvert p_{\mathrm{i}\lambda/\hbar}(\alpha_{w})\rvert\geq C\lvert w\rvert^{2}$ also holds for the finitely many words $\smash{w\in\tilde{W}}$ with $\lvert w\rvert<M$. Consequently $\lvert p_{\mathrm{i}\lambda/\hbar}(\alpha_{w})\rvert\geq C\lvert w\rvert^{2}$ holds for all words $\smash{w\in\tilde{W}}$. Setting $C_{p}\coloneqq 1/C$, the corollary follows by rearranging. $\boxempty$

We have now collected all the results needed to prove Subsection 3.1.

Proof of Subsection 3.1::

First, we note that it suffices to prove the existence of a constant $M$ such that for any multiindices $I,J\in\mathbbm{N}_{0}^{m}$ we have {\mathopen{}\mathclose{{}\left|\kern-1.07639pt\mathopen{}\mathclose{{}\left|\kern-1.07639pt\mathopen{}\mathclose{{}\left|b_{I}^{*}*^{\prime}_{\hbar}b_{J}^{*}}\right|\kern-1.07639pt}\right|\kern-1.07639pt}\right|}_{R}\leq(RM)^{\lvert I\rvert+\lvert J\rvert}. Indeed, this statement implies the continuity of $*^{\prime}_{\hbar}$ since for $p=\sum_{I\in\mathbbm{N}_{0}^{m}}p_{I}b_{I}^{*}$ and $q=\sum_{I\in\mathbbm{N}_{0}^{m}}q_{I}b_{I}^{*}$ we estimate

[TABLE]

Using the notation $I_{\{j\}}$ introduced in the proof of Subsection 2.4 we estimate

[TABLE]

The sum $\sum_{w_{(1)},\dots,w_{(\lvert I\rvert)}}$ introduced in (1) is over all partitions of $w\in\tilde{W}$ into words $w_{(1)},\dots,w_{(\lvert I\rvert)}$. To be more precise, consider a partition $P_{1},\dots,P_{\lvert I\rvert}$ of $\{1,\dots,\lvert w\rvert\}$ into $\lvert I\rvert$ many subsets. If $P_{i}=\{p_{i,1},\dots,p_{i,j_{i}}\}$ with $p_{i,1}<\dots<p_{i,j_{i}}$, then associate the word $w_{(i)}=w_{p_{i,1}}w_{p_{i,2}}\dots w_{p_{i,j_{i}}}$. Then we sum over all partitions. The other sum is defined similarly. We also used submultiplicativity of {\mathopen{}\mathclose{{}\left|\kern-1.07639pt\mathopen{}\mathclose{{}\left|\kern-1.07639pt\mathopen{}\mathclose{{}\left|{}\cdot{}}\right|\kern-1.07639pt}\right|\kern-1.07639pt}\right|}_{R} in this step. To justify (2), we note that for any $Z\in\mathfrak{gl}_{N}(\mathbbm{C})$, $Z^{\mathrm{left},(1,0)}b^{*}_{i}$ is of degree 1, so that $X_{w_{(\ell)}}^{\mathrm{left},(1,0)}b^{*}_{I_{\{\ell\}}}$ is of degree 1. Defining C\coloneqq\max_{i\in\{1,\dots,m\},\alpha\in\Delta}{\mathopen{}\mathclose{{}\left|\kern-1.07639pt\mathopen{}\mathclose{{}\left|\kern-1.07639pt\mathopen{}\mathclose{{}\left|X^{\mathrm{left},(1,0)}_{\alpha}b^{*}_{i}}\right|\kern-1.07639pt}\right|\kern-1.07639pt}\right|}_{1} we obtain

[TABLE]

The sum over $w_{(1)},\dots,w_{(\lvert I\rvert)}$ has $\lvert I\rvert^{\lvert w\rvert}$ many terms, since for each letter of $w$ we can choose in which of the $\lvert I\rvert$ many sets we want to have it. The same holds true for the other sum. In (3) we used that there are at most $\smash{k^{\lvert w\rvert}}$ many words of a given length $\lvert w\rvert$ in $\smash{\tilde{W}}$ and (4) holds, because we just added some positive extra terms.

Remark \theremark.

For a fixed compact set $K\subseteq\mathbbm{C}\setminus P_{\lambda}$ the proof above shows that there is a constant $M\in\mathbbm{R}^{+}$ such that for any $\hbar\in K$ we have

[TABLE]

since Subsection 3.1 gives uniform estimates for all $\hbar\in K$.

3.2 Stein manifolds and extension of holomorphic functions

In this subsection, we discuss extension properties of holomorphic functions on closed complex submanifolds of Stein manifolds or, more generally, on analytic subsets of Stein manifolds. We will use the results in the next subsection to identify the reduction-topology with the topology of locally uniform convergence and to determine the completion of the space of polynomials with respect to this topology.

Since analytic subsets in a Stein manifold are a very natural setting to prove the extendability results, we formulate them in this generality (even though we only need the case of closed submanifolds most of the time). The content of this subsection has been known for long and can be found e.g. in the textbook [22].

Recall that for a complex manifold $M$, we denote the vector space of holomorphic functions on $M$ by $\mathrm{Hol}(M)$.

Definition \thedefinition (Holomorphic convex hull).

For a compact subset $K$ of a complex manifold $M$ we define its holomorphic convex hull to be the set

[TABLE]

Definition \thedefinition (Stein manifold).

A complex manifold $M$ of dimension $n$ is said to be Stein if

1. i.)

for any compact subset $K\subseteq M$ its holomorphic convex hull $\hat{K}_{M}$ is compact, 2. ii.)

for every $z\in M$ there are functions $f_{1},\dots,f_{n}\in\mathrm{Hol}(M)$ that form a coordinate system around $z$.

Stein manifolds should be thought of as domains of holomorphicity for holomorphic functions of several complex variables. Clearly $\mathbbm{C}^{n}$ is Stein.

Definition \thedefinition.

A subset $V\subseteq M$ of a complex manifold is called analytic, if for every point $z\in M$ there is a neighbourhood $U\subseteq M$ of $z$ such that there is a family of holomorphic functions $f_{j}\in\mathrm{Hol}(U)$ indexed by $j$ in some index set $J$, such that

[TABLE]

Example \theexample.

Any closed complex submanifold $M$ of $\mathbbm{C}^{n}$ is an analytic subset. Indeed, around any $z\in M$ we can find a submanifold chart, that is a neighbourhood $U$ and coordinates $z=(z_{1},\dots,z_{n})$ such that $M\cap U$ is given by the vanishing of the first $n-\dim M$ coordinates. Therefore we can take $f_{j}=z_{j}$ for $j=1,\dots,n-\dim M$ in Subsection 3.2. Around any $z\notin M$ there is a neighbourhood $U$ such that $U\cap M=\emptyset$ and we may pick $f_{1}=1$ in Subsection 3.2.

Definition \thedefinition.

*A function $f\colon V\to\mathbbm{C}$ on an analytic subset $V\subseteq M$ of a complex manifold is called holomorphic, if for every point $z\in V$ there is a neighbourhood $U\subseteq M$ of $z$ and a holomorphic function $g\in\mathrm{Hol}(U)$ such that g\big{|}_{U\cap V}=f\big{|}_{U\cap V}. *

Example \theexample.

If $V$ is a closed complex submanifold of $\mathbbm{C}^{n}$ as in Subsection 3.2, then this definition of a holomorphic function coincides with the usual definition. Indeed, in any submanifold chart $(U,z)$ as in Subsection 3.2, a holomorphic function on $U\cap V$ can be extended constantly along the first $n-\dim M$ variables to a holomorphic function on $U$. The reverse implication is clear.

Proposition \theproposition.

*Let $V$ be an analytic subset of a Stein manifold $M$. Then $\mathrm{Hol}(V)$ endowed with the topology of locally uniform convergence is a Fréchet space. *

Proof**:**

It follows from the definition of analytic subsets that $V$ is closed. Therefore the restriction of any compact exhaustion of $M$ to $V$ gives a compact exhaustion $K_{i}$ of $V$. The seminorms $\lVert f\rVert_{K_{i}}=\sup_{K_{i}}\lvert f\rvert$ define a countable system of seminorms inducing the topology of locally uniform convergence. The completeness of $\mathrm{Hol}(V)$ with respect to this topology is a non-trivial result and proved in [22, Theorem 7.4.9]. $\boxempty$

The crucial property of an analytic subset $V$ of a Stein manifold is the following extendability property for any holomorphic function on $V$, see [22, Theorem 7.4.8]:

Theorem \thethm (Extendability of holomorphic functions).

*Let $V$ be an analytic subset of a Stein manifold $M$. Any holomorphic function $f\in\mathrm{Hol}(V)$ can be extended to a holomorphic function $f\in\mathrm{Hol}(M)$. In other words, the restriction map $\mathrm{Hol}(M)\to\mathrm{Hol}(V)$ is surjective. *

For an analytic subset $V$ of a complex manifold $M$ we denote the subspace of $\mathrm{Hol}(M)$ consisting of functions that vanish on $V$ by $\mathcal{I}(V)$. Note that the restriction map $\mathrm{Hol}(M)\to\mathrm{Hol}(V)$ descends to a map on the quotient, $r\colon\mathrm{Hol}(M)/\mathcal{I}(V)\to\mathrm{Hol}(V)$. This map is clearly injective by definition of $\mathcal{I}(V)$, and if $M$ is Stein it is surjective by the previous theorem.

Corollary \thecorollary.

*Assume that $M$ is Stein and that $V\subseteq M$ is an analytic subset. If $\mathrm{Hol}(M)/\mathcal{I}(V)$ is endowed with the quotient topology of the topology of locally uniform convergence and $\mathrm{Hol}(V)$ is endowed with the topology of locally uniform convergence then the map $r\colon\mathrm{Hol}(M)/\mathcal{I}(V)\to\mathrm{Hol}(V)$ is a homeomorphism. *

Proof**:**

We know that $r$ is bijective, so it only remains to prove the continuity of $r$ and $r^{-1}$. Both $\mathrm{Hol}(M)$ and $\mathrm{Hol}(V)$ are Fréchet spaces (for $\mathrm{Hol}(M)$ this is well-known, for $\mathrm{Hol}(V)$ it is the statement of Subsection 3.2). Since $\mathcal{I}(V)$ is closed, $\mathrm{Hol}(M)/\mathcal{I}(V)$ is also a Fréchet space. Clearly the locally uniform convergence of a sequence $f_{i}\in\mathrm{Hol}(M)$ implies the locally uniform convergence of the sequence of restrictions f_{i}\big{|}_{V}\in\mathrm{Hol}(V), so the map $r$ is continuous. The statement then follows from the open mapping theorem for Fréchet spaces. $\boxempty$

3.3 Characterizing the reduction-topology

In this subsection, we show that the reduction-topology on $\mathcal{O}_{\lambda}$ as defined in Subsection 3.1 is the topology of locally uniform convergence and that the completion of the space of polynomials $\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$ on $\mathcal{O}_{\lambda}$ with respect to this topology is exactly the space of holomorphic functions $\mathrm{Hol}(\mathcal{O}_{\lambda})$ on $\mathcal{O}_{\lambda}$.

Proposition \theproposition.

*The reduction topology $\mathcal{T}_{\mathrm{red}}$ on $\mathcal{O}_{\lambda}$ coincides with the topology of locally uniform convergence. *

Proof**:**

By the assumption at the beginning of this section (see also Subsection 1.1), $G$ is a closed complex submanifold of $\mathbbm{C}^{N\times N}$, hence an analytic subset by Subsection 3.2. Applying Subsection 3.2 yields that the quotient topology on $\mathrm{Hol}(G)$ induced by the topology of locally uniform convergence on $\mathbbm{C}^{N\times N}$ is precisely the topology of locally uniform convergence on $G$.

By Subsection 3.1 the reduction-topology is the restriction of this topology to the subspace of right $G_{\lambda}$-invariant holomorphic functions. Using that this subspace is closed, and that a sequence $f_{i}\in\mathrm{Hol}(\mathcal{O}_{\lambda})$ converges locally uniformly if and only if the sequence $\pi^{*}(f_{i})\in\mathrm{Hol}(G)^{G_{\lambda}}$ converges locally uniformly, one can easily check that the reduction-topology coincides with the topology of locally uniform convergence on $\mathrm{Hol}(\mathcal{O}_{\lambda})$. $\boxempty$

Finally, we would like to determine the completion $\widehat{\operatorname{\mathrm{Pol}}}(\mathcal{O}_{\lambda})$ of $\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$ with respect to the topology of locally uniform convergence.

Proposition \theproposition.

*We have $\widehat{\operatorname{\mathrm{Pol}}}(\mathcal{O}_{\lambda})=\mathrm{Hol}(\mathcal{O}_{\lambda})$. *

Proof**:**

The inclusion $\widehat{\operatorname{\mathrm{Pol}}}(\mathcal{O}_{\lambda})\subseteq\mathrm{Hol}(\mathcal{O}_{\lambda})$ is trivial, since $\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})\subseteq\mathrm{Hol}(\mathcal{O}_{\lambda})$ and the limit of a locally uniformly convergent sequence of holomorphic functions is again holomorphic.

The other inclusion is easy to see if one uses that semisimple coadjoint orbits are affine algebraic varieties, see Subsection 1.1: In particular they are analytic subsets of the Stein manifold $\mathfrak{g}^{*}$ and therefore we can use Subsection 3.2 to extend any $f\in\mathrm{Hol}(\mathcal{O}_{\lambda})$ to a holomorphic function $\tilde{f}\in\mathrm{Hol}(\mathfrak{g}^{*})$, which can be approximated by polynomials. Restricting these approximating polynomials to $\mathcal{O}_{\lambda}$ gives a sequence of polynomials in $\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$ converging locally uniformly to $f$.

Alternatively, we know that $G$ is a closed submanifold of the Stein manifold $\mathbbm{C}^{N\times N}$, so the same argument yields that any $f\in\mathrm{Hol}(G)$ can be approximated by some $p_{n}\in\operatorname{\mathrm{Pol}}(G)$. Assume that $f\in\mathrm{Hol}(G)^{G_{\lambda}}$. Let $K_{\lambda}$ be a maximal compact subgroup of $G_{\lambda}$. Averaging $p_{n}$ over $K_{\lambda}$ gives a sequence $p^{\prime}_{n}\in\operatorname{\mathrm{Pol}}(G)^{K_{\lambda}}$ that converges locally uniformly to $f$. Now $p^{\prime}_{n}$ is even $G_{\lambda}$-invariant since the action of $G$ is holomorphic, so $\pi_{*}p^{\prime}_{n}\in\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$ converges to $\pi_{*}f\in\mathrm{Hol}(\mathcal{O}_{\lambda})$. $\boxempty$

We are now able to prove the main theorem stated in the introduction to this section.

Proof of Section 3::

From Subsection 3.1 we know that the product $*_{\hbar}$ is continuous with respect to the reduction-topology. We showed in Subsection 3.3 that the reduction-topology coincides with the topology of locally uniform convergence on $\mathcal{O}_{\lambda}$. The previous proposition shows that the completion of $\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$ in this topology is $\mathrm{Hol}(\mathcal{O}_{\lambda})$. Finally $G$-invariance of the product on the completion is clear since the action of $G$ on $\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$ is continuous with respect to the topology of locally uniform convergence.

We close this section by the following proposition, which asserts that the dependence of $*_{\hbar}$ on $\hbar$ is holomorphic.

Proposition \theproposition (Holomorphic dependence on $\hbar$).

*For two fixed holomorphic functions $p,q\in\mathrm{Hol}(\mathcal{O}_{\lambda})$ and $x\in\mathcal{O}_{\lambda}$ the map $\mathbbm{C}\setminus P_{\lambda}\to\mathbbm{C}$, $\hbar\mapsto p\mathbin{*_{\hbar}}q(x)$ is holomorphic. *

Proof**:**

By construction of $\mathbin{*_{\hbar}}$ in Section 2, the map $\mathbbm{C}\setminus P_{\lambda}\to\mathbbm{C}$, $\hbar\mapsto p^{\prime}\mathbin{*_{\hbar}}q^{\prime}(x)$ is rational for $p^{\prime},q^{\prime}\in\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$. Assume that $p_{n}$, $q_{n}$ are sequences of polynomials on $\mathcal{O}_{\lambda}$ such that $p_{n}\to p$ and $q_{n}\to q$ locally uniformly. Since the estimates of Subsection 3.1 are locally uniform in $\hbar$, see Subsection 3.1, it follows that $p_{n}\mathbin{*_{\hbar}}q_{n}\to p\mathbin{*_{\hbar}}q$ locally uniformly in $\hbar$. But clearly the evaluation at $x$ is continuous, so that $\hbar\mapsto p\mathbin{*_{\hbar}}q(x)$ is a locally uniform limit of rational functions and therefore holomorphic. $\boxempty$

4 Quantizing real coadjoint orbits

We have seen in the previous sections how to construct (formal and strict) quantizations of complex coadjoint orbits. In this section, we will use these results to obtain (formal and strict) quantizations of real coadjoint orbits.

In Subsection 4.1 and Subsection 4.2 we collect some preliminary results on the complexification of a real coadjoint orbit $\mathcal{O}_{\lambda}$ and a real Lie group $G$. We define a certain class of analytic functions that we denote by $\mathcal{A}(\mathcal{O}_{\lambda})$ and $\mathcal{A}(G)$. In Subsection 4.3 we construct a quantization of real orbits by restricting the quantization of a complexification. We discuss the examples of complex projective spaces and hyperbolic discs in Subsection 4.4. Finally, we show that point evaluation functionals are positive for certain coadjoint orbits in Subsection 4.5 and compare the quantum algebras obtained for coadjoint orbits of real Lie groups with the same complexification in Subsection 4.6. Most results in the later subsections follow almost directly from the results in the complex case.

From now on, all complex Lie groups and Lie algebras will be denoted with a hat and letters without decoration will be used to denote real objects. We will also use hats for maps between complex objects, e.g. we rename the map defined in (1.8) to $\hat{\Psi}$.

4.1 Complexification

In this subsection we define the complexification of a real coadjoint orbit $\mathcal{O}_{\lambda}$ and a real Lie group $G$, and show how they are related.

For a real Lie algebra $\mathfrak{g}$, denote the space of real-valued real-linear functionals on $\mathfrak{g}$ by $\mathfrak{g}^{*}$. As before, $\hat{\mathfrak{g}}^{*}$ denotes the space of complex-valued complex-linear functionals on a complex Lie algebra $\hat{\mathfrak{g}}$. In the following, we will always assume that $\hat{\mathfrak{g}}=\mathfrak{g}\mathbin{\otimes}\mathbbm{C}$ is the complexification of $\mathfrak{g}$. In this case, any element of $\mathfrak{g}^{*}$ has a unique extension to an element of $\hat{\mathfrak{g}}^{*}$. We will perform this extension implicitly whenever necessary, without mentioning it. For example, in the following proposition, the coadjoint orbit $\hat{\mathcal{O}}_{\lambda}$ is really the coadjoint orbit through the extension of $\lambda\in\mathfrak{g}^{*}$ to an element of $\hat{\mathfrak{g}}^{*}$.

Proposition \theproposition.

*Let $\mathcal{O}_{\lambda}\subseteq\mathfrak{g}^{*}$ be a coadjoint orbit of a real connected Lie group, and assume that $\hat{\mathfrak{g}}$ is the complexification of $\mathfrak{g}$. Then $\mathcal{O}_{\lambda}$ is a submanifold of a unique complex coadjoint orbit $\hat{\mathcal{O}}_{\lambda}\subseteq\hat{\mathfrak{g}}^{*}$ of a complex connected Lie group with Lie algebra $\hat{\mathfrak{g}}$. The tangent space $\mathrm{T}_{\xi}\hat{\mathcal{O}}_{\lambda}$ of this orbit $\smash{\hat{\mathcal{O}}_{\lambda}}$ is the complexification of $\mathrm{T}_{\xi}\mathcal{O}_{\lambda}$ for every $\xi\in\mathfrak{g}^{*}$. *

Proof**:**

By Subsection 1.1 the coadjoint orbit $\mathcal{O}_{\lambda}$ is the symplectic leaf through $\lambda$ of the linear Poisson structure on $\mathfrak{g}^{*}$ defined just before Subsection 1.1. Similarly the coadjoint orbits in $\smash{\hat{\mathfrak{g}}^{*}}$ are symplectic leaves of the linear Poisson structure on $\smash{\hat{\mathfrak{g}}^{*}}$, and the symplectic leaf containing $\smash{\lambda\in\hat{\mathfrak{g}}^{*}}$ contains the whole orbit $\mathcal{O}_{\lambda}$. This proves existence and uniqueness of $\hat{\mathcal{O}}_{\lambda}$.

As in Subsection 1.1, we can identify $\mathrm{T}_{\xi}\mathcal{O}_{\lambda}$ with $\mathfrak{g}/\mathfrak{g}_{\xi}$ (as real vector spaces) and $\smash{\mathrm{T}_{\xi}\hat{\mathcal{O}}_{\lambda}}$ with $\smash{\hat{\mathfrak{g}}/\hat{\mathfrak{g}}_{\xi}}$ (as complex vector spaces) for all $\xi\in\mathcal{O}_{\lambda}$. Therefore $\smash{\mathrm{T}_{\xi}\hat{\mathcal{O}}_{\lambda}}$ is indeed the complexification of $\mathrm{T}_{\xi}\mathcal{O}_{\lambda}$. $\boxempty$

We refer to the complex coadjoint orbit $\hat{\mathcal{O}}_{\lambda}$ of the previous proposition as the complexification of $\mathcal{O}_{\lambda}$. We will show how to realize it explicitly as the coadjoint orbit of some Lie group $\hat{G}$.

Definition \thedefinition.

*Let $G$ be a real Lie group. A complexification of $G$ is a complex connected Lie group $\hat{G}$ together with an embedding $\iota\colon G\to\hat{G}$, such that the corresponding Lie algebra $\hat{\mathfrak{g}}$ is isomorphic to the complexification $\mathfrak{g}\mathbin{\otimes}\mathbbm{C}$ of $\mathfrak{g}$ and such that the map $\mathrm{T}_{e}\iota\colon\mathfrak{g}\to\hat{\mathfrak{g}}$ corresponds to the injection $X\mapsto X\mathbin{\otimes}1$ under this isomorphism. *

Note that a complexification according to this definition may fail to exist or may not be unique, if it exists. See the paragraph after Subsection 4.2 for an example of a Lie group with non-unique complexification. For a connected semisimple Lie group $G$ a complexification exists if and only if the group can be realized as a linear group: Existence for linear Lie groups is shown below, and the reverse implication follows since semisimplicity of $G$ implies semisimplicity of the complexification and complex connected semisimple Lie groups are always matrix Lie groups, see Subsection 1.1. There is a different notion of a universal complexification that does always exist, but that does not enjoy the property that $\hat{\mathfrak{g}}\cong\mathfrak{g}\mathbin{\otimes}\mathbbm{C}$. We will not use universal complexifications in this paper.

Proposition \theproposition.

*If $G$ is a real connected closed linear Lie group, then it admits a complexification $\hat{G}$. *

Proof**:**

We may assume that both $G$ and its Lie algebra $\mathfrak{g}$ are realized by real matrices. Then the complexification $\hat{\mathfrak{g}}=\mathfrak{g}\mathbin{\otimes}\mathbbm{C}$ is a Lie subalgebra of $\mathfrak{gl}_{N}(\mathbbm{C})$. We can use the exponential map to construct an immersed complex Lie subgroup $\hat{G}$ of $\mathrm{GL}_{N}(\mathbbm{C})$ containing $G$ as a subgroup and having $\hat{\mathfrak{g}}$ as Lie algebra, see e.g. [21, Chapter 5.9]. Since $G$ is a closed subgroup of $\mathrm{GL}_{N}(\mathbbm{C})$, it is also a closed subgroup of $\hat{G}$. $\boxempty$

Note that we did not claim that $\hat{G}$ is a closed subgroup of $\mathrm{GL}_{N}(\mathbbm{C})$. For semisimple Lie groups this follows automatically from Subsection 1.1.

Lemma 4.1**.**

*Let $G$ be a real connected Lie group with complexification $\hat{G}$ and let $\mathcal{O}_{\lambda}$ be a coadjoint orbit of $G$ with complexification $\hat{\mathcal{O}}_{\lambda}$. Then $\hat{\mathcal{O}}_{\lambda}$ is a coadjoint orbit of $\hat{G}$ and the embedding $\iota\colon G\to\hat{G}$ descends to an embedding $\mathcal{O}_{\lambda}\cong G/G_{\lambda}\to\hat{G}/\hat{G}_{\lambda}\cong\hat{\mathcal{O}}_{\lambda}$. *

Proof**:**

Since $\hat{G}$ is connected and has the Lie algebra $\hat{\mathfrak{g}}$, it follows from Subsection 1.1 that its coadjoint orbit through $\lambda$ is $\smash{\hat{\mathcal{O}}_{\lambda}}$. We identify $G$ with a subgroup of $\hat{G}$. Since the coadjoint action of $\hat{G}$ on $\hat{\mathfrak{g}}$ is holomorphic, $\hat{G}_{\lambda}$ is a complexification of $G_{\lambda}=\hat{G}_{\lambda}\cap G$. So the map $\iota$ descends to a map $\mathcal{O}_{\lambda}\cong G/G_{\lambda}\to\hat{G}/\hat{G}_{\lambda}\cong\smash{\hat{\mathcal{O}}_{\lambda}}$ that is still injective. To see that it is an embedding, note that the actions of $G_{\lambda}$ and $\hat{G}_{\lambda}$ on $\hat{G}$ are proper and free, so $\hat{G}$ is a principal $G_{\lambda}$ resp. $\hat{G}_{\lambda}$ bundle over $\hat{G}/G_{\lambda}$ resp. $\hat{G}/\hat{G}_{\lambda}$. This implies first that $G/G_{\lambda}\to\hat{G}/G_{\lambda}$ is still an embedding, and then that $G/G_{\lambda}\to\hat{G}/\hat{G}_{\lambda}$ also is. $\boxempty$

4.2 Polynomials and analytic functions

In this subsection we introduce polynomials $\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$ and a certain class of analytic functions $\mathcal{A}(\mathcal{O}_{\lambda})$ on a real coadjoint orbit $\mathcal{O}_{\lambda}$. $\mathcal{A}(\mathcal{O}_{\lambda})$ consists of restrictions of holomorphic functions on the complexification. In analogy to the complex case, $\mathcal{A}(\mathcal{O}_{\lambda})$ is the completion of $\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$ with respect to some locally convex topology.

All our polynomials are complex-valued. So for a real finite dimensional vector space $V$ we define $\operatorname{\mathrm{Pol}}(V)$ to be the unital complex subalgebra of $\mathscr{C}^{\infty}(V)$ generated by the linear maps. (Remember that $\mathscr{C}^{\infty}(V)$ consists of smooth functions $V\to\mathbbm{C}$.) So $\operatorname{\mathrm{Pol}}(V)\cong\mathrm{S}(V^{*}_{\mathbbm{C}})$ where $V^{*}_{\mathbbm{C}}$ is the complexification of V^{*}=\{\phi\colon V\to\mathbbm{R},\text{ \phi linear}\}.

Definition \thedefinition (Polynomials).

Let $\mathcal{O}_{\lambda}$ be a coadjoint orbit of a real connected Lie group $G$ with Lie algebra $\mathfrak{g}$. Then

[TABLE]

*is called the algebra of polynomials on $\mathcal{O}_{\lambda}$. *

Note that polynomials on a complex orbit $\hat{\mathcal{O}}_{\lambda}$ were assumed to be holomorphic and do therefore not coincide with polynomials on the underlying real orbit. We will always use holomorphic polynomials on complexifications, so this will hopefully not cause any confusion.

Denote the ideal of polynomials on $\mathfrak{g}^{*}$ resp. $\hat{\mathfrak{g}}^{*}$ vanishing on $\mathcal{O}_{\lambda}$ resp. $\hat{\mathcal{O}}_{\lambda}$ by $\mathcal{I}(\mathcal{O}_{\lambda})$ resp. $\mathcal{I}(\hat{\mathcal{O}}_{\lambda})$. It is clear that the maps $\operatorname{\mathrm{Pol}}(\mathfrak{g}^{*})/\mathcal{I}(\mathcal{O}_{\lambda})\to\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$ and $\operatorname{\mathrm{Pol}}(\hat{\mathfrak{g}}^{*})/\mathcal{I}(\hat{\mathcal{O}}_{\lambda})\to\operatorname{\mathrm{Pol}}(\hat{\mathcal{O}}_{\lambda})$ are isomorphisms. We would now like to relate polynomials on $\mathcal{O}_{\lambda}$ and $\hat{\mathcal{O}}_{\lambda}$.

Proposition \theproposition.

*Let $\mathcal{O}_{\lambda}\subseteq\mathfrak{g}^{*}$ be a real coadjoint orbit with complexification $\hat{\mathcal{O}}_{\lambda}\subseteq\hat{\mathfrak{g}}^{*}$. Then the restriction map \smash{({}\cdot{})\big{|}_{\mathcal{O}_{\lambda}}}\colon\mathscr{C}^{\infty}(\hat{\mathcal{O}}_{\lambda})\to\mathscr{C}^{\infty}(\mathcal{O}_{\lambda}) restricts to an isomorphism \smash{({}\cdot{})\big{|}_{\mathcal{O}_{\lambda}}}\colon\operatorname{\mathrm{Pol}}(\hat{\mathcal{O}}_{\lambda})\to\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda}). *

Proof**:**

Since restriction to $V$ is a bijection between complex linear maps $V\mathbin{\otimes}\mathbbm{C}\to\mathbbm{C}$ and real linear maps $V\to\mathbbm{C}$ for any finite dimensional real vector space $V$, it follows that the restriction map $\operatorname{\mathrm{Pol}}(\hat{\mathfrak{g}}^{*})\to\operatorname{\mathrm{Pol}}(\mathfrak{g}^{*})$ is an isomorphism. If we can prove that the restriction map $\mathcal{I}(\hat{\mathcal{O}}_{\lambda})\to\mathcal{I}(\mathcal{O}_{\lambda})$ is also an isomorphism, then we are done since $\operatorname{\mathrm{Pol}}(\hat{\mathcal{O}}_{\lambda})\cong\operatorname{\mathrm{Pol}}(\hat{\mathfrak{g}}^{*})/\mathcal{I}(\hat{\mathcal{O}}_{\lambda})\to\operatorname{\mathrm{Pol}}(\mathfrak{g}^{*})/\mathcal{I}(\mathcal{O}_{\lambda})\cong\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$ would be an isomorphism.

Since any map vanishing on $\hat{\mathcal{O}}_{\lambda}$ vanishes in particular on $\mathcal{O}_{\lambda}\subseteq\hat{\mathcal{O}}_{\lambda}$, the restriction map $\mathcal{I}(\hat{\mathcal{O}}_{\lambda})\to\mathcal{I}(\mathcal{O}_{\lambda})$ is well-defined and it is injective since it is the restriction of the injective map $\operatorname{\mathrm{Pol}}(\hat{\mathfrak{g}}^{*})\to\operatorname{\mathrm{Pol}}(\mathfrak{g}^{*})$. So we only need to prove surjectivity, meaning that if a polynomial $p$ on $\mathfrak{g}^{*}$ vanishes on $\mathcal{O}_{\lambda}$, then its unique extension to a polynomial $\hat{p}$ on $\hat{\mathfrak{g}}^{*}$ vanishes on $\hat{\mathcal{O}}_{\lambda}$. Since $\hat{\mathcal{O}}_{\lambda}$ is a complex submanifold of $\hat{\mathfrak{g}}^{*}$, the restriction of $\hat{p}$ to $\hat{\mathcal{O}}_{\lambda}$ is holomorphic. As such it is determined by its derivatives (of all orders) at $\lambda$. It is even determined by its derivatives in the direction of $\mathrm{T}_{\lambda}\mathcal{O}_{\lambda}$ since $\mathrm{T}_{\lambda}\hat{\mathcal{O}}_{\lambda}$ is the complexification of $\mathrm{T}_{\lambda}\mathcal{O}_{\lambda}$. But all these derivatives vanish since the restriction of $\hat{p}$ to $\mathcal{O}_{\lambda}$ vanishes. $\boxempty$

Definition \thedefinition.

*Let $G$ be a linear real Lie group. Its algebra of polynomials $\operatorname{\mathrm{Pol}}(G)$ is the unital complex subalgebra of $\mathscr{C}^{\infty}(G)$ generated by the functions $P_{ij}\colon G\to\mathbbm{C}$, $g\mapsto g_{ij}$. *

In contrast to the complex case, the algebra of polynomials $\operatorname{\mathrm{Pol}}(G)$ may depend on the way in which $G$ is realized as a linear group, even in the semisimple case. We will give an instructive example after stating the following proposition, which can be proven in much the same way as Subsection 4.2.

Proposition \theproposition.

*Let $G\subseteq\mathrm{GL}_{N}(\mathbbm{R})$ be a linear connected Lie group with complexification $\hat{G}\subseteq\mathrm{GL}_{N}(\mathbbm{C})$. Then the restriction map \smash{({}\cdot{})\big{|}_{G}}\colon\mathscr{C}^{\infty}(\hat{G})\to\mathscr{C}^{\infty}(G) restricts to an isomorphism \smash{({}\cdot{})\big{|}_{G}}\colon\operatorname{\mathrm{Pol}}(\hat{G})\to\operatorname{\mathrm{Pol}}(G). *

The reason why the algebra of polynomials $\operatorname{\mathrm{Pol}}(G)$ may depend on the linear structure of $G$, is essentially that $G$ may not have a unique complexification. Consider the linear semisimple Lie group $\mathrm{SL}_{3}(\mathbbm{R})\subseteq\mathrm{GL}_{3}(\mathbbm{R})$, which has $\mathrm{SL}_{3}(\mathbbm{C})$ as a complexification. The images of $\mathrm{SL}_{3}(\mathbbm{R})$ and $\mathrm{SL}_{3}(\mathbbm{C})$ under $\operatorname{\mathrm{Ad}}$ are again semisimple Lie groups. Furthermore, $\operatorname{\mathrm{Ad}}(\mathrm{SL}_{3}(\mathbbm{R}))\cong\mathrm{SL}_{3}(\mathbbm{R})$ since $\mathrm{SL}(3,\mathbbm{R})$ has trivial center, and $\operatorname{\mathrm{Ad}}(\mathrm{SL}_{3}(\mathbbm{C}))\cong\mathrm{SL}_{3}(\mathbbm{C})/\{1,\mathrm{e}^{2\pi\mathrm{i}/3},\mathrm{e}^{4\pi\mathrm{i}/3}\}$ is a complexification of $\operatorname{\mathrm{Ad}}(\mathrm{SL}_{3}(\mathbbm{R}))$. By the previous proposition $\operatorname{\mathrm{Pol}}(\operatorname{\mathrm{Ad}}(\mathrm{SL}_{3}(\mathbbm{R})))\cong\operatorname{\mathrm{Pol}}(\mathrm{SL}_{3}(\mathbbm{C})/\{1,\mathrm{e}^{2\pi\mathrm{i}/3},\mathrm{e}^{4\pi\mathrm{i}/3}\})\to\operatorname{\mathrm{Pol}}(\mathrm{SL}_{3}(\mathbbm{C}))\cong\operatorname{\mathrm{Pol}}(\mathrm{SL}_{3}(\mathbbm{R}))$ where the map in the middle is not surjective, since there are polynomials on $\mathrm{SL}_{3}(\mathbbm{C})$ that are not constant on $\{1,\mathrm{e}^{2\pi\mathrm{i}/3},\mathrm{e}^{4\pi\mathrm{i}/3}\}$.

We denote the inverses of the isomorphisms in Subsection 4.2 and Subsection 4.2 by

[TABLE]

Lemma 4.2**.**

*Let $G$ be a real connected linear Lie group with complexification $\hat{G}$, and let $\lambda\in\mathfrak{g}^{*}$ be such that $\hat{G}_{\lambda}$ is connected. If $f\in\operatorname{\mathrm{Pol}}(\hat{G})$ satisfies f\big{|}_{G}\in\operatorname{\mathrm{Pol}}(G)^{G_{\lambda}} then $f\in\smash{\operatorname{\mathrm{Pol}}(\hat{G})^{\hat{G}_{\lambda}}}$. *

Proof**:**

Let $f$ be as in the statement of the lemma. Since f\big{|}_{G}=(g\mathbin{\triangleright}f)\big{|}_{G} holds for all $g\in G_{\lambda}$ it follows from the injectivity of \smash{({}\cdot{})\big{|}_{G}} that $f=g\mathbin{\triangleright}f$, so $f\in\smash{\operatorname{\mathrm{Pol}}(\hat{G})^{G_{\lambda}}}$. Therefore $f$ is in particular invariant under $\mathfrak{g}_{\lambda}$, thus also under $\hat{\mathfrak{g}}_{\lambda}$ since the action is holomorphic. Since $\hat{G}_{\lambda}$ is connected we obtain that $f$ is $\hat{G}_{\lambda}$-invariant. $\boxempty$

Corollary \thecorollary.

*Let $G$ be a real connected semisimple linear Lie group with complexification $\hat{G}$, and assume that $\lambda\in\mathfrak{g}^{*}$ is semisimple. In this case the restriction map \smash{({}\cdot{})\big{|}_{G}}\colon\smash{\operatorname{\mathrm{Pol}}(\hat{G})^{\hat{G}_{\lambda}}}\to\operatorname{\mathrm{Pol}}(G)^{G_{\lambda}} is an isomorphism. *

Proof**:**

Note that $G_{\lambda}$ is connected by Subsection 1.1. So this is an immediate consequence of Subsection 4.2 and Lemma 4.2. $\boxempty$

Corollary \thecorollary.

*Let $G$ be a real connected semisimple linear Lie group with complexification $\hat{G}$, and assume that $\lambda\in\mathfrak{g}^{*}$ is semisimple. Then the map $\pi^{*}\colon\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})\to\operatorname{\mathrm{Pol}}(G)^{G_{\lambda}}$ is an isomorphism. *

Proof**:**

The composition $\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})\xrightarrow{\hat{{}\cdot{}}}\operatorname{\mathrm{Pol}}(\hat{\mathcal{O}}_{\lambda})\xrightarrow{\hat{\pi}^{*}}\operatorname{\mathrm{Pol}}(\hat{G})^{\hat{G}_{\lambda}}\xrightarrow{({}\cdot{})\mid_{G}}\operatorname{\mathrm{Pol}}(G)^{G_{\lambda}}$ equals $\pi^{*}$ and is an isomorphism because of Subsection 4.2, Subsection 2.3, and Subsection 4.2. $\boxempty$

Corollary \thecorollary.

Let $G$ be a real connected semisimple linear Lie group with complexification $\hat{G}$, and assume that $\lambda\in\mathfrak{g}^{*}$ is semisimple. Then the following diagram commutes and all arrows are isomorphisms:

[TABLE]

Next, we want to introduce a class of analytic functions, that becomes the closure of the polynomials with respect to a certain locally convex topology. To this end, assume that $\mathcal{O}_{\lambda}$ is a coadjoint orbit with complexification $\hat{\mathcal{O}}_{\lambda}$, and that $G$ is a real connected Lie group with complexification $\hat{G}$. Then define

[TABLE]

Note that an element $f\in\mathcal{A}(\mathcal{O}_{\lambda})$ determines a unique element $\hat{f}\in\mathrm{Hol}(\hat{\mathcal{O}}_{\lambda})$: Existence follows by definition of $\mathcal{A}(\mathcal{O}_{\lambda})$ and $\hat{f}$ is determined by all its derivatives at $\lambda$. Since the complexification of $\mathrm{T}_{\lambda}\mathcal{O}_{\lambda}$ is just $\mathrm{T}_{\lambda}\hat{\mathcal{O}}_{\lambda}$, see Lemma 4.1, it suffices to take derivatives in the direction of $\mathrm{T}_{\lambda}\mathcal{O}_{\lambda}$. But these derivatives are determined by $f$. A similar reasoning holds for $G$ and $\hat{G}$. We obtain a commuting square that is similar to the square for polynomials obtained in Subsection 4.2.

Proposition \theproposition.

The following diagram is commutative and all arrows are isomorphisms:

[TABLE]

Proof**:**

We know from Subsection 1.1 that $\hat{\pi}^{*}\colon\mathrm{Hol}(\hat{\mathcal{O}}_{\lambda})\to\mathrm{Hol}(\hat{G})^{\hat{G}_{\lambda}}$ is an isomorphism. In the previous paragraph we explained that $\hat{}\cdot{}\colon\mathcal{A}(\mathcal{O}_{\lambda})\to\smash{\mathrm{Hol}(\hat{\mathcal{O}}_{\lambda})}$ and $\hat{}\cdot{}\colon\mathcal{A}(G)\to\mathrm{Hol}(\hat{G})$ are isomorphisms and as in Lemma 4.2 it follows that the same is true for $\hat{}\cdot{}\colon\mathcal{A}(G)^{G_{\lambda}}\to\smash{\mathrm{Hol}(\hat{G})^{\hat{G}_{\lambda}}}$. Composing these isomorphisms we obtain that $\pi^{*}\colon\mathcal{A}(\mathcal{O}_{\lambda})\to\mathcal{A}(G)^{G_{\lambda}}$ is an isomorphism. $\boxempty$

Since $\operatorname{\mathrm{Pol}}(\hat{\mathcal{O}}_{\lambda})\subseteq\mathrm{Hol}(\hat{\mathcal{O}}_{\lambda})$ it follows that $\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})\subseteq\mathcal{A}(\mathcal{O}_{\lambda})$. We can define a topology $\smash{\mathcal{T}_{\widehat{\mathrm{lu}}}}$ of extended locally uniform convergence on $\mathcal{A}(\mathcal{O}_{\lambda})$ as follows: A sequence $f_{n}\in\mathcal{A}(\mathcal{O}_{\lambda})$ converges to some $f\in\mathcal{A}(\mathcal{O}_{\lambda})$ if and only if the sequence $\smash{\hat{f}_{n}}\in\smash{\mathrm{Hol}(\hat{\mathcal{O}}_{\lambda})}$ converges locally uniformly to $\hat{f}\in\mathrm{Hol}(\hat{\mathcal{O}}_{\lambda})$. Clearly the maps $\hat{}\cdot{}\colon\mathcal{A}(\mathcal{O}_{\lambda})\to\mathrm{Hol}(\hat{\mathcal{O}}_{\lambda})$ and ({}\cdot{})\big{|}_{\mathcal{O}_{\lambda}}\colon\mathrm{Hol}(\hat{\mathcal{O}}_{\lambda})\to\mathcal{A}(\mathcal{O}_{\lambda}) are both homeomorphisms. From Subsection 3.3 it follows that the closure of $\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$ with respect to the topology of extended locally uniform convergence is $\mathcal{A}(\mathcal{O}_{\lambda})$.

4.3 Formal and strict star products on real coadjoint orbits

In a sense all constructions in Section 1, Section 2, and Section 3 are compatible with the restriction to real forms. In this subsection we want to make this statement precise. In particular, we will show that we can restrict formal and strict products from a complexification $\hat{\mathcal{O}}_{\lambda}$ of a semisimple coadjoint orbit $\mathcal{O}_{\lambda}$ of a real connected semisimple Lie group $G$ to formal and strict star products on $\mathcal{O}_{\lambda}$. These star products can—as before—be computed by applying fundamental vector fields or by passing to the Lie group by using the maps $\pi^{*}$ and $\pi_{*}$. We will determine when the star products on $\mathcal{O}_{\lambda}$ are of (pseudo) Wick type or of standard ordered type.

Proposition \theproposition.

Let $\mathcal{O}_{\lambda}$ be a semisimple coadjoint orbit of a semisimple connected real Lie group $G$. By Lemma 4.1 it has a complexification $\hat{\mathcal{O}}_{\lambda}$, and for $\hbar\in\mathbbm{C}\setminus P_{\lambda}$ there are strict products $\hat{*}_{\hbar}\colon\operatorname{\mathrm{Pol}}(\hat{\mathcal{O}}_{\lambda})\times\operatorname{\mathrm{Pol}}(\hat{\mathcal{O}}_{\lambda})\to\operatorname{\mathrm{Pol}}(\hat{\mathcal{O}}_{\lambda})$ with extensions $\hat{*}_{\hbar}\colon\mathrm{Hol}(\hat{\mathcal{O}}_{\lambda})\times\mathrm{Hol}(\hat{\mathcal{O}}_{\lambda})\to\mathrm{Hol}(\hat{\mathcal{O}}_{\lambda})$ constructed in Subsection 2.3 and Section 3. These products restrict to $G$-invariant strict products

[TABLE]

*for all $\hbar\in\mathbbm{C}\setminus P_{\lambda}$. For fixed $p,q\in\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$, the dependence of $p*_{\hbar}q$ on $\hbar$ is rational with no pole at zero, and for fixed $f,g\in\mathcal{A}(\mathcal{O}_{\lambda})$ and $x\in\mathcal{O}_{\lambda}$, the dependence of $f*_{\hbar}g(x)$ on $\hbar$ is holomorphic. Both products are continuous with respect to the topology of extended locally uniform convergence defined at the end of Subsection 4.2. *

Proof**:**

Since the restriction maps $\operatorname{\mathrm{Pol}}(\hat{\mathcal{O}}_{\lambda})\to\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$ and $\mathrm{Hol}(\hat{\mathcal{O}}_{\lambda})\to\mathcal{A}(\mathcal{O}_{\lambda})$ are both homeomorphisms (with respect to the topology of locally uniform convergence on the domains and the topology of extended locally uniform convergence on the codomains), the statement follows trivially from the corresponding statements for $\hat{*}_{\hbar}$, obtained in Subsection 2.3, Section 3, and Subsection 3.3. $\boxempty$

We would like to compute these star products without passing to the complexification. The construction of bidifferential operators from Subsection 1.2 works completely similarly in the real setting. Recall that our differential operators act on complex-valued functions, and therefore any complex vector field $\Gamma^{\infty}(\mathrm{T}^{\mathbbm{C}}M)$ defines a first order differential operator on $M$.

Proposition \theproposition.

Let $G$ be a real Lie group with Lie algebra $\mathfrak{g}$, and let $\hat{\mathfrak{g}}$ be the complexification of $\mathfrak{g}$. The map

[TABLE]

obtained by extending $\hat{\mathfrak{g}}\ni X\mapsto X^{\mathrm{left}}\in\Gamma^{\infty}(\mathrm{T}^{\mathbbm{C}}G)$ to an algebra homomorphism $\mathscr{U}\hat{\mathfrak{g}}\to\operatorname{\mathrm{DiffOp}}^{G}(G)$ and further to tensor products as in (1.5c) is an isomorphism. If $H$ is a closed Lie subgroup of $G$, then the map

[TABLE]

*is also an isomorphism. *

Proof**:**

With the obvious modifications the proofs of Subsection 1.2 and Subsection 1.2 given in Appendix A.1 apply also to the real situation. $\boxempty$

To be consistent with the notation of this chapter, we denote the map defined in (1.8) by $\hat{\Psi}$.

Lemma 4.3**.**

Let $G$ be a real Lie group with closed subgroup $H$ and assume that the complex Lie group $\hat{G}$ is a complexification of $G$ and contains a complex closed subgroup $\hat{H}$ that is a complexification of $H$. The maps $({}\cdot{})^{\mathrm{left}}$ and $\Psi$ are compatible with the maps $({}\cdot{})^{\mathrm{left},(1,0)}$ and $\hat{\Psi}$ in the sense that the diagrams

[TABLE]

*commute for all open subsets $\hat{U}\subseteq\hat{G}$ and $\hat{V}\subseteq\hat{G}/\hat{H}$, with $U\coloneqq\hat{U}\cap G$ and $V\coloneqq\hat{V}\cap G/H$, and all elements $\vec{u}\in(\mathscr{U}\hat{\mathfrak{g}})^{\mathbin{\otimes}k}$ and $\vec{v}\in((\mathscr{U}\hat{\mathfrak{g}}/\mathscr{U}\hat{\mathfrak{g}}\cdot\hat{\mathfrak{h}})^{\mathbin{\otimes}k})^{\hat{H}}$. *

Proof**:**

The commutativity of the second diagram follows easily from commutativity of the first, since the restrictions are compatible with $\pi^{*}$ and $\pi_{*}$. To prove commutativity of the first diagram, assume that $k=1$ and $\vec{u}=X\in\hat{\mathfrak{g}}\subseteq\mathscr{U}\hat{\mathfrak{g}}$. The tangent map of a holomorphic function commutes with the multiplication by $\mathrm{i}$. We compute

[TABLE]

for $f\in\mathrm{Hol}(\hat{U})$ and $g\in U$. The general case follows from this computation by the way in which $({}\cdot{})^{\mathrm{left},(1,0)}$ and $({}\cdot{})^{\mathrm{left}}$ are extended to $(\mathscr{U}\hat{\mathfrak{g}})^{\mathbin{\otimes}k}$. $\boxempty$

Corollary \thecorollary.

Let $\mathcal{O}_{\lambda}$ be a semisimple coadjoint orbit of a semisimple connected real Lie group $G$. For $\hbar\in\mathbbm{C}\setminus P_{\lambda}$ and $p,q\in\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$, the product $*_{\hbar}\colon\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})\times\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})\to\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$ defined in Subsection 4.3 can be computed by

[TABLE]

Proof**:**

The previous lemma implies

[TABLE]

Note that the sum over $\ell$ is finite by Subsection 2.3. $\boxempty$

Theorem \thethm.

*Let $\mathcal{O}_{\lambda}$ be a semisimple coadjoint orbit of a semisimple connected real Lie group $G$. The product $\star\colon\mathscr{C}^{\infty}(\mathcal{O}_{\lambda})[\!\!\;[\hbar]\!\!\;]\times\mathscr{C}^{\infty}(\mathcal{O}_{\lambda})[\!\!\;[\hbar]\!\!\;]\to\mathscr{C}^{\infty}(\mathcal{O}_{\lambda})[\!\!\;[\hbar]\!\!\;]$ defined by $f\star g=\Psi(F)(f,g)$ where $F$ was obtained in Subsection 2.3 is a $G$-invariant formal star product. In particular, it is associative and deforms the KKS symplectic form on $\mathcal{O}_{\lambda}$. Furthermore, $p\star q$ coincides with the formal power series expansion of $p*_{\hbar}q$ around $\hbar=0$ for $p,q\in\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$, and f\star g=\hat{f}\mathbin{\hat{\star}}\hat{g}\big{|}_{\mathcal{O}_{\lambda}} for $f,g\in\mathcal{A}(\mathcal{O}_{\lambda})$. *

Proof**:**

It is immediate from the definition of $F$ and $\Psi$ that every order of $\star$ is given by a $G$-invariant bidifferential operator. Since $F$ is the formal power series expansion of $F_{\hbar}$ around $\hbar=0$ and $p*_{\hbar}q$ is rational with no pole at [math] for $p,q\in\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$, it follows that $p\star q$ coincides with the formal power series expansion of $p*_{\hbar}q$. The compatibility with $\mathbin{\hat{\star}}$ is immediate from Lemma 4.3. Since bidifferential operators are uniquely determined by their behaviour on $\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})\subseteq\mathcal{A}(\mathcal{O}_{\lambda})$, the compatibility with $\hat{\star}$ implies that $\star$ is associative and, using Subsection 2.3, that it deforms the KKS symplectic form. $\boxempty$

Recall that we proved in Subsection 2.3 that the product $\hat{*}_{\hbar}$ separates variables with respect to the distributions $L_{+}$ and $L_{-}$, which we call $\hat{L}_{+}$ and $\hat{L}_{-}$ in this section. In the real case, those distributions may have further properties. They can be real, or the holomorphic and antiholomorphic tangent spaces with respect to a complex structure. Before giving further details let us make the following definitions.

Definition \thedefinition (Star products of standard ordered type).

*A star product $*_{\hbar}$ on a symplectic manifold $M$ is said to be of standard ordered type if there are two Lagrangian distributions $L_{1},L_{2}\subseteq\mathrm{T}M$ spanning the real tangent bundle $\mathrm{T}M$ of $M$ such that the first argument of the star product is derived only in directions of $L_{1}$ and the second argument only in directions of $L_{2}$. *

Definition \thedefinition (Star products of (pseudo) Wick type).

*A star product $*_{\hbar}$ on a complex manifold $M$ that is also symplectic is said to be of pseudo Wick type if the first argument is derived only in holomorphic directions and the second argument only in antiholomorphic directions. A star product of pseudo Wick type on a Kähler manifold is said to be of Wick type. *

For formal star products of Wick type and with respect to the usual ∗-involution given by complex conjugation, point evaluations are positive linear functionals, which is not necessarily the case for formal star products of pseudo Wick type. Note that the situation is more complicated for strict star products, as we shall see in Subsection 4.5.

Let us briefly recall some results on the existence of invariant complex structures on coadjoint orbits. See Appendix A.3 for more details. Let $\mathcal{O}_{\lambda}$ be a semisimple coadjoint orbit of a real connected semisimple Lie group $G$ with Lie algebra $\mathfrak{g}$, and assume that $G_{\lambda}$ is compact. Choose a real Cartan subalgebra $\mathfrak{h}$ containing $\lambda^{\sharp}$. Since $\mathfrak{h}\subseteq\mathfrak{g}_{\lambda}$, it follows that $\mathfrak{h}$ is compact (meaning that it integrates to a subgroup of $G$ with compact closure). Then there are $G$-invariant complex structures on $\mathcal{O}_{\lambda}$, and these structures are in bijection to invariant orderings of $\hat{\Delta}$ (we say an ordering on $\hat{\Delta}$ is invariant if it is the restriction of an invariant ordering of $\Delta$ as defined in Subsection 2.2) as follows. Recall that $\mathrm{T}_{\lambda}^{\mathbbm{C}}\mathcal{O}_{\lambda}\cong\hat{\mathfrak{g}}/\hat{\mathfrak{g}}_{\lambda}\cong\bigoplus_{\alpha\in\hat{\Delta}}\mathfrak{g}^{\alpha}$. So given an invariant ordering we can define a map $I_{\lambda}\colon\mathrm{T}_{\lambda}^{\mathbbm{C}}\mathcal{O}_{\lambda}\to\mathrm{T}_{\lambda}^{\mathbbm{C}}\mathcal{O}_{\lambda}$ by letting $I_{\lambda}X_{\alpha}=\mathrm{i}X_{\alpha}$ if $\alpha\in\smash{\hat{\Delta}^{+}}$, and $I_{\lambda}X_{\alpha}=-\mathrm{i}X_{\alpha}$ if $\alpha\in\smash{\hat{\Delta}^{-}}$. The map $I_{\lambda}$ extends $G$-invariantly to an endomorphism $I$ of the complexified tangent bundle $\mathrm{T}^{\mathbbm{C}}\mathcal{O}_{\lambda}$ and restricts to an endomorphism of the real tangent bundle $\mathrm{T}\mathcal{O}_{\lambda}$, thus it defines a complex structure.

If $G$ is compact, there is a unique ordering that makes $\mathcal{O}_{\lambda}$ with the complex structure $I$ and the KKS symplectic form $\omega_{\mathrm{KKS}}$ a Kähler manifold. This ordering is characterized by $\alpha\in\smash{\hat{\Delta}}$ being positive iff $(\alpha,\mathrm{i}\lambda)>0$. In particular it is standard. See Appendix A.3 for more details.

Proposition \theproposition.

For a semisimple coadjoint orbit $\mathcal{O}_{\lambda}$ of a real connected semisimple linear Lie group $G$, the product $*_{\hbar}$ obtained in Subsection 4.3

1. i.)

has poles $P_{\lambda}\subseteq\mathbbm{R}$ if $\mathfrak{h}$ is compact, 2. ii.)

is of pseudo Wick type if $G_{\lambda}$ is compact and the same ordering is used in the construction of the star product and the definition of the complex structure, 3. iii.)

is of standard ordered type with poles $P_{\lambda}\subseteq\mathrm{i}\mathbbm{R}$ if $\mathrm{i}\mathfrak{h}\subseteq\hat{\mathfrak{g}}$ is compact.

*In particular, if $G$ is compact and, in the construction of $*_{\hbar}$, one chooses the ordering that makes $\mathcal{O}_{\lambda}$ with the induced complex structure $I$ a Kähler manifold, then $*_{\hbar}$ is of Wick type. *

Proof**:**

Roots take purely imaginary values on a compact Lie subalgebra of $\mathfrak{h}$. Since $\lambda\in\mathfrak{g}^{*}$ is by definition real on $\mathfrak{h}\subseteq\smash{\hat{\mathfrak{h}}}$, it follows that $(\lambda,\mu)\in\mathrm{i}\mathbbm{R}$ if $\mathfrak{h}$ is compact and $(\lambda,\mu)\in\mathbbm{R}$ if $\mathrm{i}\mathfrak{h}$ is compact. Since $\frac{1}{2}(\mu,\mu)-(\rho,\mu)\in\mathbbm{R}$, this implies that the roots (with respect to $\hbar$) of $p_{\mathrm{i}\lambda/\hbar}(\mu)=\frac{1}{2}(\mu,\mu)-(\rho,\mu)-\frac{\mathrm{i}}{\hbar}(\lambda,\mu)$ are real if $\mathfrak{h}$ is compact and purely imaginary if $\mathrm{i}\mathfrak{h}$ is compact.

Recall the definition of the distributions $L_{+}$ and $L_{-}$, which we denote by $\hat{L}_{+}$ and $\hat{L}_{-}$ in this section, made just after Lemma 2.14. Restricting them to $\mathcal{O}_{\lambda}\subseteq\hat{\mathcal{O}}_{\lambda}$ gives two distributions $L_{+},L_{-}\subseteq\mathrm{T}^{\mathbbm{C}}\mathcal{O}_{\lambda}$ of the complexified tangent bundle. An analogue of Subsection 1.2 in the real case and the explicit formula for $F_{\hbar}$ from Subsection 2.2 together with Subsection 2.3 show that $\star$ derives the first argument only in directions of $L_{+}$, and the second argument only in directions of $L_{-}$.

Assume that $\mathfrak{g}_{\lambda}$ is compact. The holomorphic tangent space $\smash{\mathrm{T}_{\lambda}^{(1,0)}\mathcal{O}_{\lambda}}$ is, under the isomorphism $\smash{\mathrm{T}_{\lambda}^{\mathbbm{C}}\mathcal{O}_{\lambda}}\cong\smash{\hat{\mathfrak{g}}/\hat{\mathfrak{g}}_{\lambda}}$, spanned by $X_{\alpha}-\mathrm{i}I_{\lambda}X_{\alpha}$ for $\alpha\in\hat{\Delta}$. If $I_{\lambda}$ is defined using the ordering chosen in the construction of $*_{\hbar}$ as described above, then $X_{\alpha}-\mathrm{i}I_{\lambda}X_{\alpha}=X_{\alpha}-\mathrm{i}\cdot\mathrm{i}X_{\alpha}=2X_{\alpha}$ if $\alpha\in\hat{\Delta}^{+}$, and $X_{\alpha}-\mathrm{i}I_{\lambda}X_{\alpha}=X_{\alpha}-\mathrm{i}\cdot(-\mathrm{i})X_{\alpha}=0$ if $\alpha\in\hat{\Delta}^{-}$, so \smash{\mathrm{T}_{\lambda}^{(1,0)}\mathcal{O}_{\lambda}}=\smash{\operatorname{\mathrm{span}}\{(X_{\alpha})_{\mathcal{O}_{\lambda}}\big{|}{}_{\lambda},\alpha\in\hat{\Delta}^{+}\}}. This coincides exactly with \smash{L_{+}\big{|}_{\lambda}}, and by $G$-invariance it follows that $L_{+}$ coincides with $\mathrm{T}^{(1,0)}\mathcal{O}_{\lambda}$. Similarly, $L_{-}$ coincides with $\mathrm{T}^{(0,1)}\mathcal{O}_{\lambda}$. Therefore $\star$ is of pseudo Wick type.

If $\mathrm{i}\mathfrak{h}$ is compact, then every $\operatorname{\mathrm{ad}}_{H}$ for $H\in\mathfrak{h}$ is self-adjoint. Since they are all commuting we can find simultaneous eigenvectors in $\mathfrak{g}$ of all $\operatorname{\mathrm{ad}}_{H}$ (without complexifying $\mathfrak{g}$). But then we can pick $X_{\alpha}$ and $Y_{\alpha}$ to lie in $\mathfrak{g}$ so that $L_{1}=L_{+}\cap\mathfrak{g}$ and $L_{2}=L_{-}\cap\mathfrak{g}$ are Lagrangian distributions satisfying Subsection 4.3. $\boxempty$

Remark \theremark.

Assume that $\mathfrak{g}_{\lambda}$ is compact as in part ii*.)* of the previous proposition. If one uses different invariant orderings in the construction of the star product and in the definition of a complex structure, then the distributions $L_{+}$ and $L_{-}$ may both contain holomorphic and antiholomorphic directions. Since we are mainly interested in star products of (pseudo) Wick type (these are the ones for which we would hope to find positive linear functionals on the star product algebra, see Subsection 4.5), we will usually assume that the two orderings agree.

4.4 Examples: complex projective spaces and hyperbolic discs

Recall that we have computed the canonical element of the Shapovalov pairing for $\mathrm{SL}_{1+n}(\mathbbm{C})$ and a certain choice of $\lambda$ in Subsection 2.4. Let us now specialize this result to the real forms $\mathrm{SU}(1+n)$ and $\mathrm{SU}(1,n)$.

Example \theexample ($\mathbbm{CP}^{n}$).

The coadjoint orbit of $\mathrm{SU}(1+n)$ through $\lambda\colon\mathfrak{su}_{1+n}\to\mathbbm{R}$, $X\mapsto-\mathrm{i}rX_{0,0}$ with $r\in\mathbbm{R}^{+}$ is the complex projective space $\mathbbm{CP}^{n}$. $\mathrm{SL}_{1+n}(\mathbbm{C})$ is a complexification of $\mathrm{SU}(1+n)$. Using the notation $\hat{\mathfrak{h}}$ for the Cartan subalgebra of $\mathfrak{sl}_{1+n}(\mathbbm{C})$ introduced in Subsection 2.4, we obtain a compact Cartan subalgebra $\mathfrak{h}\coloneqq\mathfrak{su}_{1+n}\cap\hat{\mathfrak{h}}$ of $\mathfrak{su}_{1+n}$. Subsection A.3 tells us that the Kähler complex structure is defined by the ordering of $\hat{\Delta}$ for which $\alpha\in\hat{\Delta}^{+}$ iff $(\mathrm{i}\lambda,\alpha)>0$. This ordering is the restriction of the ordering on $\Delta$ for which all $\alpha_{i,j}$ with $i<j$ are positive. Therefore the element $F_{\hbar}$ from Subsection 2.4 induces a Wick type star product on $\mathbbm{CP}^{n}$. This product has poles at $\{\frac{1}{n}r\mid n\in\mathbbm{N}\}$.

Example \theexample ($\mathbbm{D}^{n}$).

Denote the complex hyperbolic disc in $n$ dimensions by $\mathbbm{D}^{n}$. Recall that $\mathrm{SU}(1,n)$ denotes the group of isometries of the indefinite scalar product $g(v,w)=-v_{0}w_{0}+\sum_{i=1}^{n}v_{i}w_{i}$ on $\mathbbm{R}^{1+n}$. The coadjoint orbit of $\mathrm{SU}(1,n)$ through $\lambda\colon\mathfrak{su}_{1,n}\to\mathbbm{R}$, $X\mapsto-\mathrm{i}rX_{0,0}$ with $r\in\mathbbm{R}^{+}$ is the hyperbolic disc $\mathbbm{D}^{n}$. $\mathrm{SL}_{1+n}(\mathbbm{C})$ is a complexification of $\mathrm{SU}(1,n)$. Again, $\mathfrak{h}\coloneqq\mathfrak{su}_{1,n}\cap\hat{\mathfrak{h}}$ defines a compact Cartan subalgebra of $\mathfrak{su}_{1,n}$. Now all roots are non-compact, so that according to Subsection A.3 the Kähler complex structure is defined by the ordering on $\hat{\Delta}$ for which $\alpha\in\hat{\Delta}^{+}$ iff $(\mathrm{i}\lambda,\alpha)<0$. This ordering is the restriction of the ordering on $\Delta$ for which all $\alpha_{i,j}$ with $i>j$ are positive. Therefore the element $F_{\hbar}$ from Subsection 2.4 induces a Wick type star product on $\mathbbm{D}^{n}$. This product has poles at $\{-\frac{1}{n}r\mid n\in\mathbbm{N}\}$.

Remark \theremark.

A star product of Wick type on the hyperbolic disc was also studied in [29], where it was obtained from a star product of Wick type on $\mathbbm{C}^{1+n}$ using phase space reduction. This product coincides with the star product obtained in Subsection 4.4. To see this, one checks that monomials of degree 1 generate the star product algebra, so that it suffices to compare the two formulas for a degree 1 monomial and an arbitrary monomial. But for a degree 1 monomial only very few summands are non-zero in both constructions and one can explicitly check that the expressions agree.

4.5 Positive linear functionals

In this subsection we prove that for certain coadjoint orbits and certain values of $\hbar$ the point evaluation functionals of the star product algebras constructed in Subsection 4.3 are positive. In order to have a meaningful notion of positivity we need a star involution on $(\mathcal{A}(\mathcal{O}_{\lambda}),*_{\hbar})$. Of course, this star involution should be the restriction of the complex conjugation of $\mathscr{C}^{\infty}(\mathcal{O}_{\lambda})$, but we need to prove that this restriction is well-defined.

Assume that $\hat{\mathfrak{g}}=\mathfrak{g}\mathbin{\otimes}\mathbbm{C}$ is the complexification of a Lie algebra $\mathfrak{g}$. The complex conjugation $\overline{{}\cdot{}\vphantom{\lambda}}\colon\hat{\mathfrak{g}}\to\hat{\mathfrak{g}}$, $X\mathbin{\otimes}z\mapsto X\mathbin{\otimes}\overline{z}$ is an antilinear involution on $\hat{\mathfrak{g}}$. Then $\overline{{}\cdot{}\vphantom{\lambda}}\colon\hat{\mathfrak{g}}^{*}\to\hat{\mathfrak{g}}^{*}$, $\phi\mapsto\overline{\phi}\coloneqq\overline{{}\cdot{}\vphantom{\lambda}}{}\circ\phi\circ{}\overline{{}\cdot{}\vphantom{\lambda}}$ defines an antilinear involution on $\hat{\mathfrak{g}}^{*}$. Note that on the right hand side, we first apply the involution of $\hat{\mathfrak{g}}$, then $\phi$, and then the complex conjugation of $\mathbbm{C}$. Therefore the right hand side defines a complex linear functional $\overline{\phi}\in\hat{\mathfrak{g}}^{*}$. The map $\phi\mapsto\overline{\phi}$ is antilinear.

Lemma 4.4**.**

*Let $G\subseteq\mathrm{GL}_{N}(\mathbbm{R})$ be a real linear Lie group with complexification $\hat{G}\subseteq\mathrm{GL}_{N}(\mathbbm{C})$, assume $\lambda\in\mathfrak{g}^{*}$, and let $\hat{\mathcal{O}}_{\lambda}$ be the coadjoint orbit of $\hat{G}$ through $\lambda$. Then the map $\overline{{}\cdot{}\vphantom{\lambda}}\colon\hat{\mathfrak{g}}^{*}\to\hat{\mathfrak{g}}^{*}$ restricts to an antilinear involution $\overline{{}\cdot{}\vphantom{\lambda}}\colon\hat{\mathcal{O}}_{\lambda}\to\hat{\mathcal{O}}_{\lambda}$. *

Proof**:**

Note that since $\lambda\in\mathfrak{g}^{*}$ we have $\overline{\lambda}=\lambda$. Therefore we compute

[TABLE]

Here $\overline{g}$ denotes the entrywise complex conjugate of $g\in\hat{G}$. Since the exponential map $\hat{\mathfrak{g}}\to\hat{G}$ commutes with the complex conjugation, it follows that $\hat{G}$ is closed under entrywise complex conjugation, and therefore $\overline{g}\in\hat{G}$ and $\smash{\operatorname{\mathrm{Ad}}_{\overline{g}}^{*}\lambda}\in\hat{\mathcal{O}}_{\lambda}$. This proves that $\overline{{}\cdot{}\vphantom{\lambda}}$ restricts to $\hat{\mathcal{O}}_{\lambda}$, and the restriction is clearly still an antilinear involution. $\boxempty$

Note that $\mathrm{T}_{\xi}\overline{{}\cdot{}\vphantom{\lambda}}\circ I_{\xi}=(I_{\overline{\xi}})^{-1}\circ\mathrm{T}_{\xi}\overline{{}\cdot{}\vphantom{\lambda}}$ holds for $\xi\in\hat{\mathfrak{g}}^{*}$, where $\mathrm{T}_{\xi}\overline{{}\cdot{}\vphantom{\lambda}}\colon\mathrm{T}_{\xi}\hat{\mathfrak{g}}^{*}\to\mathrm{T}_{\overline{\xi}}\hat{\mathfrak{g}}^{*}$ is the tangent map to the complex conjugation of $\hat{\mathfrak{g}}^{*}$ and $I_{\xi}\colon\mathrm{T}_{\xi}\hat{\mathfrak{g}}^{*}\to\mathrm{T}_{\xi}\hat{\mathfrak{g}}^{*}$ is the complex structure at $\xi$. Since the complex structure $I$ and the complex conjugation $\overline{{}\cdot{}\vphantom{\lambda}}$ of $\mathcal{O}_{\lambda}$ are both obtained by restriction from $\hat{\mathfrak{g}}^{*}$, they satisfy the same relation.

For any $f\in\mathrm{Hol}(\hat{\mathcal{O}}_{\lambda})$ consider the function $f^{*}\coloneqq\overline{{}\cdot{}\vphantom{\lambda}}\circ f\circ\overline{{}\cdot{}\vphantom{\lambda}}$, where the left $\overline{{}\cdot{}\vphantom{\lambda}}$ is the complex conjugation of $\mathbbm{C}$ and the right $\overline{{}\cdot{}\vphantom{\lambda}}$ is the antilinear involution obtained in the previous lemma. Denote the complex structure of $\mathbbm{C}$ by $J$, and identify the tangent space of $\mathbbm{C}$ with $\mathbbm{C}$. Then

[TABLE]

shows that $f^{*}$ is holomorphic. Since $\overline{{}\cdot{}\vphantom{\lambda}}$ restricts to the identity on $\mathcal{O}_{\lambda}\subseteq\mathfrak{g}^{*}$, it follows that $f^{*}|_{\mathcal{O}_{\lambda}}=\overline{f|_{\mathcal{O}_{\lambda}}}$. Consequently, the restriction of ${}^{*}\colon\mathrm{Hol}(\hat{\mathcal{O}}_{\lambda})\to\mathrm{Hol}(\hat{\mathcal{O}}_{\lambda})$ to $\mathcal{A}(\mathcal{O}_{\lambda})$ is just the complex conjugation $\smash{\overline{{}\cdot{}\vphantom{\lambda}}}\colon\mathcal{A}(\mathcal{O}_{\lambda})\to\mathcal{A}(\mathcal{O}_{\lambda})$. In other words, the complex conjugation is well-defined on $\mathcal{A}(\mathcal{O}_{\lambda})$.

Proposition \theproposition.

*Let $\mathcal{O}_{\lambda}$ be a semisimple coadjoint orbit of a connected semisimple real Lie group $G$. Assume that the Cartan subalgebra $\mathfrak{h}$ used in the construction of a star product $*_{\hbar}$ is compact. Then $\overline{f*_{\hbar}g}=\overline{g}*_{\overline{\hbar}}\overline{f}$ holds for all $f,g\in\mathcal{A}(\mathcal{O}_{\lambda})$. *

Proof**:**

As in the proof of Subsection 4.3 one argues that since $\mathfrak{h}$ is compact the coefficients $\smash{p_{\mathrm{i}\lambda}^{w}(\alpha_{w})}$ are real and more generally $\smash{\overline{p_{\mathrm{i}\lambda/\hbar}^{w}(\alpha_{w})}}=\smash{p_{\mathrm{i}\lambda/\smash{\overline{\hbar}}}^{w}(\alpha_{w})}$. From (A.3) we obtain that $\smash{\overline{X_{\alpha}\mathbin{\otimes}Y_{\alpha}}}=Y_{\alpha}\mathbin{\otimes}X_{\alpha}=\tau(X_{\alpha}\mathbin{\otimes}Y_{\alpha})$ for both a compact and a non-compact root $\smash{\alpha\in\hat{\Delta}^{+}}$, and the same formula holds when $\alpha$ is replaced by a word $w\in\tilde{W}$. Here $\smash{\overline{{}\cdot{}\vphantom{\lambda}}}$ is the complex conjugation of $\hat{\mathfrak{g}}$ with respect to $\mathfrak{g}$, extended to $(\mathscr{U}\hat{\mathfrak{g}})^{\mathbin{\otimes}2}$, and $\tau\colon(\mathscr{U}\hat{\mathfrak{g}})^{\mathbin{\otimes}2}\to(\mathscr{U}\hat{\mathfrak{g}})^{\mathbin{\otimes}2}$ is the flip of the two tensor factors. Note that $\tau$ stays well-defined on $(\mathscr{U}\hat{\mathfrak{g}}/\mathscr{U}\hat{\mathfrak{g}}\cdot\hat{\mathfrak{g}}_{\lambda})^{\mathbin{\otimes}2}$, and therefore the formula for $F_{\hbar}$ obtained in Subsection 2.2, Subsection 2.3, and the computations above imply $\overline{F_{\hbar,\ell}}=\tau(F_{\overline{\hbar},\ell})$. Consequently

[TABLE]

holds for all $f,g\in\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$ and extends to $\mathcal{A}(\mathcal{O}_{\lambda})$ by continuity. $\boxempty$

A linear functional $\phi$ on a ∗-algebra $\mathscr{A}$ is said to be positive if $\phi(a^{*}a)\geq 0$ for all $a\in\mathscr{A}$. In the following we formulate our results for the star algebra $\mathscr{A}_{\hbar}\coloneqq(\mathcal{A}(\mathcal{O}_{\lambda}),*_{\hbar},\overline{{}\cdot{}\vphantom{\lambda}})$, but would like to point out that they also hold for $(\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda}),*_{\hbar},\overline{{}\cdot{}\vphantom{\lambda}})$.

Theorem \thethm.

*Assume that $\mathcal{O}_{\lambda}$ is a semisimple coadjoint orbit of a real connected semisimple Lie group $G$. Assume further that $\mathfrak{h}$ is a compact Cartan subalgebra, and that all roots (with respect to the complexification $\smash{\hat{\mathfrak{h}}}$ of $\mathfrak{h}$) in $\smash{\hat{\Delta}}$ are non-compact. Let $*_{\hbar}$ be the star product constructed with respect to the ordering for which $\alpha\in\hat{\Delta}$ is positive if and only if $(\alpha,\mathrm{i}\lambda)<0$. Then there is a constant $M>0$ such that for all $\xi\in\mathcal{O}_{\lambda}$ and $\hbar\in(0,M)\setminus P_{\lambda}$ the point evaluation at $\xi$ is a positive linear functional $\mathrm{ev}_{\xi}\colon\mathscr{A}_{\hbar}\to\mathbbm{C}$. *

Proof**:**

Since $(\alpha,\mathrm{i}\lambda)<0$ for all $\alpha\in\hat{\Delta}^{+}$, it follows that $-\mathrm{i}(\lambda,\mu)>0$ holds for all $\mu\in\mathbbm{N}_{0}\hat{\Delta}^{+}\setminus\{0\}$. There are only finitely many $\mu\in\mathbbm{N}_{0}\hat{\Delta}^{+}$ with $(\rho,\mu)-\frac{1}{2}(\mu,\mu)>0$, thus we can choose $M>0$ such that $-\smash{\frac{\mathrm{i}}{\hbar}}(\lambda,\mu)>(\rho,\mu)-\smash{\frac{1}{2}(\mu,\mu)}$ holds for all $\mu\in\smash{\mathbbm{N}_{0}\hat{\Delta}^{+}}\setminus\{0\}$ and $\hbar\in(0,M)\setminus P_{\lambda}$. But this says precisely that $p_{\mathrm{i}\lambda/\hbar}(\mu)>0$, and therefore $\smash{p_{\mathrm{i}\lambda/\hbar}^{w}(\alpha_{w})}>0$ for all $\smash{w\in\tilde{W}}$. For a non-compact root we have $\smash{\overline{X_{\alpha}}}=Y_{\alpha}$ according to (A.3b). Consequently, if $g\in G$ is such that $\xi=\operatorname{\mathrm{Ad}}^{*}_{g}(\lambda)$, then

[TABLE]

holds for all $f\in\mathcal{A}(\mathcal{O}_{\lambda})$. $\boxempty$

Example \theexample ($\mathbbm{D}^{n}$).

It is straightforward to check that the choices made to quantize the hyperbolic disc in Subsection 4.4 are such that $\mathfrak{h}$ is compact, such that every root in $\hat{\Delta}$ is non-compact, and such that $\alpha\in\hat{\Delta}$ is positive iff $(\alpha,\mathrm{i}\lambda)<0$. Therefore the previous theorem implies the existence of a constant $M>0$ such that all point evaluation functionals are positive if $\hbar\in(0,M)$.

We can prove a stronger result by using the formula for $F_{\hbar}$ derived in Subsection 2.4. If $\hbar\in(0,\infty)$ then all the coefficients appearing in this formula are positive, and so point evaluations are positive for all $\hbar\in(0,\infty)$.

Note that a similar proof does not work for $\mathbbm{CP}^{n}$ since some of the coefficients in (2.39) are negative. Indeed, one can use the appearing negative coefficients to show that no point evaluation functional is positive on $\mathbbm{CP}^{n}$ for $\hbar\in(0,\infty)\setminus P_{\lambda}$.

4.6 A generalized Wick rotation

In this subsection we want to state an immediate corollary of the construction in the previous sections. Let $\mathfrak{g}_{1}$, $\mathfrak{g}_{2}$ be two real semisimple Lie algebras with the same complexification $\smash{\hat{\mathfrak{g}}}$. Assume $\lambda\in\mathfrak{g}_{1}^{*}\cap\mathfrak{g}_{2}^{*}$ where we view $\mathfrak{g}_{1}^{*}$ and $\mathfrak{g}_{2}^{*}$ as subspaces of $\hat{\mathfrak{g}}^{*}$. Denote the coadjoint orbits in $\mathfrak{g}_{1}^{*}$ and $\mathfrak{g}_{2}^{*}$ through $\lambda$ by $\mathcal{O}_{\lambda}^{1}$ and $\mathcal{O}_{\lambda}^{2}$, respectively. There is an isomorphism $\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda}^{1})\to\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda}^{2})$ given by composing the map $\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda}^{1})\ni p\mapsto\hat{p}\in\smash{\operatorname{\mathrm{Pol}}(\hat{\mathcal{O}}_{\lambda})}$ with the restriction to $\mathcal{O}_{\lambda}^{2}$. Here $\hat{\mathcal{O}}_{\lambda}$ is the complex extension of $\mathcal{O}_{\lambda}$. It turns out that this isomorphism is still an isomorphism of both the uncompleted and completed quantum algebras.

Theorem \thethm.

*Let $\mathfrak{g}_{1}$ and $\mathfrak{g}_{2}$ be two real semisimple Lie algebras with a common complexification $\hat{\mathfrak{g}}$ and assume that $\lambda\in\mathfrak{g}_{1}^{*}\cap\mathfrak{g}_{2}^{*}$ is semisimple. Then the algebras $(\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda}^{1}),*^{1}_{\hbar})$ and $(\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda}^{2}),*^{2}_{\hbar})$, and also the algebras $(\mathcal{A}(\mathcal{O}_{\lambda}^{1}),*^{1}_{\hbar})$ and $(\mathcal{A}(\mathcal{O}_{\lambda}^{2}),*^{2}_{\hbar})$, constructed with respect to the same Cartan subalgebra $\mathfrak{h}\subseteq\mathfrak{g}_{1}\cap\mathfrak{g}_{2}$ and the same ordering, are isomorphic. *

Proof**:**

Both algebras are isomorphic to $(\operatorname{\mathrm{Pol}}(\hat{\mathcal{O}}_{\lambda}),\mathbin{\hat{*}_{\hbar}})$ or $(\mathrm{Hol}(\hat{\mathcal{O}}_{\lambda}),\mathbin{\hat{*}_{\hbar}})$. $\boxempty$

Example \theexample ($\mathbbm{CP}^{n}$ and $\mathbbm{D}^{n}$).

We know from Subsection 4.4 and Subsection 4.4 that $\mathbbm{CP}^{n}$ and $\mathbbm{D}^{n}$ are coadjoint orbits of the Lie groups $\mathrm{SU}(1+n)$ and $\mathrm{SU}(1,n)$ through the same element, and that $\mathrm{SL}_{1+n}(\mathbbm{C})$ is a common complexification. So the previous proposition implies that the star product algebras on $\mathbbm{CP}^{n}$ and $\mathbbm{D}^{n}$ are isomorphic if we choose the same ordering in the construction of the star products.

The ordering that induces a Kähler complex structure on $\mathbbm{CP}^{n}$, induces the complex structure on $\mathbbm{D}^{n}$ that is the opposite of the Kähler complex structure. Therefore the associated star product on $\mathbbm{D}^{n}$ is of pseudo Wick type with respect to this opposite complex structure, and therefore of anti-Wick type for the Kähler complex structure. (A star product is of anti-Wick type if the first argument is derived in antiholomorphic directions and the second argument is derived in holomorphic ones.) Consequently, the algebra $\mathcal{A}(\mathbbm{CP}^{n})$ with the Wick type star product is isomorphic to the algebra $\mathcal{A}(\mathbbm{D}^{n})$ with the anti-Wick type star product. Similarly, the algebra $\mathcal{A}(\mathbbm{CP}^{n})$ with the anti-Wick type star product is isomorphic to the algebra $\mathcal{A}(\mathbbm{D}^{n})$ with the Wick type star product.

One can also construct an isomorphism between the Wick type star product for $\hbar$ and the anti-Wick type star product for $-\hbar$, both on the hyperbolic disc and the complex projective space. Composing with these isomorphisms shows that the Wick type star product for $\hbar$ on $\mathbbm{CP}^{n}$ is isomorphic to the Wick type star product for $-\hbar$ on $\mathbbm{D}^{n}$.

Note that Subsection 4.6 only gives an algebra homomorphism between $\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda}^{1})$ and $\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda}^{2})$, or between $\mathcal{A}(\mathcal{O}_{\lambda}^{1})$ and $\mathcal{A}(\mathcal{O}_{\lambda}^{2})$. If we view these algebras as ∗-algebras with the star involution considered in Subsection 4.5 then they are in general not ∗-isomorphic! One can see this for example by proving that the point evaluation functionals on $\mathbbm{CP}^{n}$ are not positive for $\hbar\in(0,\infty)\setminus P_{\lambda}$.

Appendix A Proofs, $G$-finite functions, and complex structures

In Appendix A.1 we prove Subsection 1.2 and Subsection 1.2. In Appendix A.2 we prove Subsection 2.3 using the concept of $G$-finite functions. Finally we recall some facts about complex structures on coadjoint orbits in Appendix A.3.

A.1 Proofs of Subsection 1.2 and

Subsection 1.2

Let $M$ be a manifold. For $f\in\mathscr{C}^{\infty}(M)$ we define $M_{f}\colon\mathscr{C}^{\infty}(M)\to\mathscr{C}^{\infty}(M)$, $f^{\prime}\mapsto ff^{\prime}$ and $\smash{M_{f}^{i}}=\mathsf{id}^{\times(i-1)}\times M_{f}\times\mathsf{id}^{\times(k-i)}\colon\mathscr{C}^{\infty}(M)^{k}\to\mathscr{C}^{\infty}(M)^{k}$.

Definition \thedefinition.

Let $M$ be a manifold. For a multiindex $K=(K_{1},\dots,K_{k})\in\mathbbm{Z}^{k}$ we define \text{k$-$\operatorname{\mathrm{DiffOp}}}_{K}(M)=\{0\} if some $K_{i}<0$ and otherwise we define inductively

[TABLE]

*Here $(K-E_{i})_{j}=K_{j}-\delta_{ij}$ where $\delta_{ij}$ is $1$ if $i=j$ and [math] otherwise. Elements of \text{k$-$\operatorname{\mathrm{DiffOp}}}_{K}(M) are called $k$-differential operators of degree $K$. A map $D\colon\mathscr{C}^{\infty}(M)^{k}\to\mathscr{C}^{\infty}(M)$ is said to be a $k$-differential operator if it is a $k$-differential operator of some degree $K$. The space of $k$-differential operators is denoted by \text{k$-$\operatorname{\mathrm{DiffOp}}}(M). *

It follows that a $k$-differential operator is local in every argument, so that it can be restricted to any open subset. In a chart $U\subseteq M$ with local coordinates $(x^{1},\dots,x^{n})$, a $k$-differential operator $D$ of degree $K$ can be written as

[TABLE]

where $c_{I_{1},\dots,I_{k}}\in\mathscr{C}^{\infty}(M)$ and $c_{I_{1},\dots,I_{k}}=0$ if $\lvert I_{i}\rvert>K_{i}$ for some $1\leq i\leq k$. For a multiindex $J\in\mathbbm{N}_{0}^{n}$ we used $\partial_{x}^{J}\coloneqq\partial_{x^{1}}^{J_{1}}\dots\partial_{x^{n}}^{J_{n}}$ and $\partial_{x^{i}}\coloneqq\frac{\partial}{\partial x^{i}}$. Conversely, an operator $D\colon\mathscr{C}^{\infty}(M)^{k}\to\mathscr{C}^{\infty}(M)$ that has this form in any chart is $k$-differential of order $K$. A $k$-differential operator $D$ on a complex manifold $M$ is holomorphic if, in local holomorphic coordinates $(z^{1},\dots,z^{n})$, we have

[TABLE]

with all $c_{I_{1},\dots,I_{k}}$ being holomorphic. Here $\partial_{z}^{J}=\partial_{z^{1}}^{J_{1}}\dots\partial_{z^{n}}^{J_{n}}$ and $\partial_{z^{i}}=\frac{\partial}{\partial z^{i}}$. Equivalently, $D$ is holomorphic if $D$ maps $\mathrm{Hol}(U)^{k}$ into $\mathrm{Hol}(U)$ and D\big{|}_{U}\circ M_{f}^{i}-M_{f}\circ D\big{|}_{U}=0 for all open subsets $U\subseteq M$ and all antiholomorphic functions $f$ on $U$. We write \text{k$-$\operatorname{\mathrm{DiffOp}}}_{\mathcal{H}}(M) for the space of holomorphic $k$-differential operators.

We say a $k$-differential operator is of order $K\in\mathbbm{Z}^{k}$ at a point $p\in M$ if, when written in a local chart $U$ around $p$ as in (A.2), we have $c_{I_{1},\dots,I_{k}}(p)=0$ whenever $\lvert I_{j}\rvert>K_{j}$ for some $1\leq j\leq k$.

If $I_{1},\dots,I_{k},J,K\in\mathbbm{N}_{0}^{n}$ are all multiindices, we write $J\leq K$ if $J_{i}\leq K_{i}$ for all $1\leq i\leq n$. If $X_{1},\dots,X_{n}\in\mathfrak{g}$, then we use $X^{J}$ as a shorthand for $X_{1}^{J_{1}}\dots X_{n}^{J_{n}}\in\mathscr{U}\mathfrak{g}$ and $X^{I_{1}\mathbin{\otimes}\dots\mathbin{\otimes}I_{k}}$ as a shorthand for $X^{I_{1}}\mathbin{\otimes}\dots\mathbin{\otimes}X^{I_{k}}\in(\mathscr{U}\mathfrak{g})^{\mathbin{\otimes}k}$.

Proof of Subsection 1.2::

Choose a basis $\{X_{1},\dots,X_{n}\}$ of $\mathfrak{g}$. It follows from the Poincaré–Birkhoff–Witt theorem that $\{X^{I_{1}\mathbin{\otimes}\dots\mathbin{\otimes}I_{k}}\mid I_{1},\dots,I_{k}\in\mathbbm{N}_{0}^{n}\}$ is a basis of $(\mathscr{U}\mathfrak{g})^{\mathbin{\otimes}k}$. Moreover, \{X_{1}^{\mathrm{left},(1,0)}\big{|}_{e},\dots,\allowbreak X_{n}^{\mathrm{left},(1,0)}\big{|}_{e}\} is a basis of the tangent space $\smash{\mathrm{T}_{e}^{(1,0)}G}$ and we can choose a complex chart $U$ around $e$ with local coordinates $(z^{1},\dots,z^{n})$ such that \partial_{z^{i}}\big{|}_{e}=X_{i}^{\mathrm{left},(1,0)}\big{|}_{e}.

Assume $\vec{u}=\sum_{I_{1},\dots,I_{k}\in\mathbbm{N}_{0}^{n}}c_{I_{1},\dots,I_{k}}X^{I_{1}\mathbin{\otimes}\dots\mathbin{\otimes}I_{k}}\neq 0$ with only finitely many $c_{I_{1},\dots,I_{k}}\neq 0$. Choose $I_{1},\dots,I_{k}$ in such a way that $c_{I_{1},\dots,I_{k}}\neq 0$ and $c_{J_{1},\dots,J_{k}}=0$ whenever $I_{i}\leq J_{i}$ and $(I_{1},\dots,I_{k})\neq(J_{1},\dots J_{k})$. For $\vec{f}=(z^{I_{1}},\dots,z^{I_{k}})\in\mathscr{C}^{\infty}(U)^{\times k}$ we compute $\vec{u}^{\mathrm{left},(1,0)}\vec{f}(e)=I_{1}!\dots I_{k}!c_{I_{1},\dots,I_{k}}\neq 0$. So $\vec{u}^{\mathrm{left},(1,0)}\neq 0$ and $({}\cdot{})^{\mathrm{left},(1,0)}$ is injective.

Note that $(X^{I_{1}})^{\mathrm{left},(1,0)}f_{1}\cdot\ldots\cdot(X^{I_{k}})^{\mathrm{left},(1,0)}f_{k}=\partial_{z}^{I_{1}}f_{1}\cdot\ldots\cdot\partial_{z}^{I_{k}}f_{k}+D^{\prime}(f_{1},\dots,f_{k})$ where $D^{\prime}$ is a holomorphic $k$-differential operator whose order at $e$ is strictly smaller than $(\lvert I_{1}\rvert,\dots,\lvert I_{k}\rvert)$. For any holomorphic $k$-differential operator $D$ we can therefore, by induction, find coefficients $c_{I_{1},\dots,I_{k}}\in\mathbbm{C}$, only finitely many of which are non-zero, such that

[TABLE]

holds for all $f_{1},\dots,f_{k}\in\mathscr{C}^{\infty}(G)$. In other words, $D$ and the left-invariant differential operator $\sum_{I_{1},\dots,I_{k}\in\mathbbm{N}_{0}^{n}}(c_{I_{1},\dots,I_{k}}X^{I_{1}\mathbin{\otimes}\dots\mathbin{\otimes}I_{k}})^{\mathrm{left},(1,0)}$ agree at $e$. So if $D$ is also left-invariant, then these operators agree everywhere on $G$, proving surjectivity.

The proof of Subsection 1.2 is similar. We need the following lemma to simplify the local calculations.

Lemma A.1**.**

Let $G$ be a complex Lie group with Lie algebra $\mathfrak{g}$, and assume that $H$ is a closed complex Lie subgroup of $G$ with Lie algebra $\mathfrak{h}$. Given a basis $B=\{X_{1},\dots,X_{n}\}$ of $\mathfrak{g}$ such that $B^{\prime}=\{X_{n-r+1},\dots,X_{n}\}$ is a basis of $\mathfrak{h}$ one can choose a neighbourhood $U$ of $e$ in $G$ and complex coordinates $z=(z^{1},\dots,z^{n})$ on $U$ such that

1. i.)

for any $g\in U$ its fiber $gH\cap U$ is given locally as $(\{z(g)\}+\{0\}\times\mathbbm{C}^{r})\cap z(U)$, 2. ii.)

the left-invariant holomorphic vector fields agree with coordinate vector fields at $e\in G$, that is X_{i}^{\mathrm{left},(1,0)}\big{|}_{e}=\partial_{z^{i}}\big{|}_{e}.

Proof**:**

It is well known that $\pi\colon G\to G/H$ is a principal bundle. Therefore we can choose a local trivialization $\chi\colon\pi^{-1}(V)\to V\times H$ on a small neighbourhood $V$ of $eH$ in $G/H$. Choosing coordinates on $V$ (after possibly shrinking $V$ first) and on a neighbourhood $W$ of the identity in $H$, we obtain coordinates $z^{\prime}$ on $U\coloneqq\chi^{-1}(V\times W)\subseteq G$ satisfying property i.). Since all $X_{i}^{\mathrm{left},(1,0)}$ are linearly independent we can write X_{i}^{\mathrm{left},(1,0)}\big{|}_{e}=A_{ij}\partial_{(z^{\prime})^{j}}\big{|}_{e} for some invertible matrix $A$ and since $X_{i}^{\mathrm{left},(1,0)}$ is tangential to $H\subseteq G$ for $i>n-r$, it follows that $A_{ij}=0$ for $i>n-r$, $j\leq n-r$. Then the coordinates $z\coloneqq(A^{-1})^{\mathrm{T}}z^{\prime}$ satisfy both properties of the lemma. $\boxempty$

Let $\pi\colon G\to G/H$. Given coordinates as in the previous lemma we may identify $\pi(U)$ locally with $\{(z^{1}(g),\dots,z^{n-r}(g),0,\dots,0)\mid g\in U\}$. Then $(z^{1},\dots,z^{n-r})$ descend to coordinates on $\pi(U)$ and $\pi$ is, with respect to these coordinates, given by the projection to the first $n-r$ coordinates.

Lemma A.2**.**

*The map $\Psi$ from Subsection 1.2 is injective. *

Proof**:**

Let $r=\dim\mathfrak{h}$ and $n=\dim\mathfrak{g}\geq r$. We can choose a basis $B=\{X_{1},\dots,X_{n}\}$ of $\mathfrak{g}$ such that $B^{\prime}=\{X_{n-r+1},\dots,X_{n}\}$ is a basis of $\mathfrak{h}$. Recall from the proof of Subsection 1.2 that $\{X^{I_{1}\mathbin{\otimes}\dots\mathbin{\otimes}I_{k}}\mid I_{1},\dots,I_{k}\in\mathbbm{N}_{0}^{n}\}$ is a basis of $(\mathscr{U}\mathfrak{g})^{\mathbin{\otimes}k}$. Furthermore,

[TABLE]

is a basis of the ideal $I$ defined just before Lemma 1.1 and

[TABLE]

is a basis of a complement $C$ of $I$ in $(\mathscr{U}\mathfrak{g})^{\mathbin{\otimes}k}$. Injectivity of $\Psi$ means that [math] is the only element of $C$ on which $\Psi$ vanishes.

So to prove that $\Psi$ is injective, it suffices to find, for any non-zero

[TABLE]

some open subset $U\subseteq G/H$ and some $k$-tuple of functions $\vec{f}\in\mathscr{C}^{\infty}(U)^{k}$ such that $\Psi([\vec{u}])(\vec{f})\neq 0$. Fix $\vec{u}\in C\setminus\{0\}$ and assume that $I_{1},\dots,I_{k}\in\mathbbm{N}_{0}^{n-r}$ are chosen such that $c_{I_{1},\dots,I_{k}}\neq 0$ and such that for any multiindices $J_{1},\dots,J_{k}\in\mathbbm{N}_{0}^{n-r}$ satisfying $I_{i}\leq J_{i}$ and $(I_{1},\dots,I_{k})\neq(J_{1},\dots,J_{k})$ we have $c_{J_{1},\dots,J_{k}}=0$. Choose coordinates $z=(z^{1},\dots,z^{n})$ around $e$ on $G$ as in the previous lemma, and note that, as described just after this lemma, $(z^{1},\dots,z^{n-r})$ descend to coordinates $(y^{1},\dots,y^{n-r})$ on $G/H$. Set $\vec{f}=(y^{I_{1}},\dots,y^{I_{k}})$, so that $\pi^{*}\vec{f}=(z^{I_{1}},\dots,z^{I_{k}})$. This implies that

[TABLE]

$\boxempty$

Lemma A.3**.**

*The map $\Psi$ from Subsection 1.2 is surjective. *

Proof**:**

We claim that for any holomorphic $k$-differential operator $D$ on $G/H$ we can find $\vec{u}\in(\mathscr{U}\mathfrak{g})^{\mathbin{\otimes}k}$ such that

[TABLE]

holds for all $\vec{f}\in\mathscr{C}^{\infty}(G/H)^{k}$. We prove this claim by induction on the order $K\in\mathbbm{Z}^{k}$ of $D$ at $eH$. If $K_{i}<0$ for some $1\leq i\leq k$, then $D=0$ and we can use $\vec{u}=0$. For the induction step, assume that the claim is already proven for every holomorphic $k$-differential operator of order strictly smaller than $K$ at $eH$. Choose coordinates $z=(z^{1},\dots,z^{n})$ around $e$ on $G$ as in Lemma A.1 and denote the coordinates on $G/H$ induced by $(z^{1},\dots,z^{n-r})$ by $y\coloneqq(y^{1},\dots,y^{n-r})$. Locally we can write

[TABLE]

with $c_{I_{1},\dots,I_{k}}\in\mathscr{C}^{\infty}(G/H)$ satisfying $c_{I_{1},\dots,I_{k}}(eH)=0$ whenever $\lvert I_{i}\rvert>K_{i}$ for some $1\leq i\leq k$. Define a holomorphic $k$-differential operator $D_{G}$ on $G$ by

[TABLE]

Then $D_{G}(\pi^{*}\vec{f})(e)=\pi^{*}(D\vec{f})(e)$. Set $\vec{u}_{1}\coloneqq\sum_{I_{1},\dots,I_{k}\in\mathbbm{N}_{0}^{n-r}}c_{I_{1},\dots,I_{k}}(\pi(e))X^{I_{1}}\mathbin{\otimes}\dots\mathbin{\otimes}X^{I_{k}}\in(\mathscr{U}\mathfrak{g})^{\mathbin{\otimes}k}$. Note that $D^{\prime}_{G}\coloneqq D_{G}-\vec{u}_{1}^{\mathrm{left},(1,0)}$ has a strictly smaller order than $D_{G}$ at $e$ since X_{i}^{\mathrm{left},(1,0)}\big{|}_{e}=\partial_{z^{i}}\big{|}_{e}. There are functions $c^{\prime}_{I_{1},\dots,I_{k}}\in\mathscr{C}^{\infty}(G)$ such that we can express $D^{\prime}_{G}$ in local coordinates as

[TABLE]

We obtain a $k$-differential operator $D^{\prime}$ on $G/H$ of strictly smaller order than $D$ at $eH$ by letting

[TABLE]

It fulfils $D^{\prime}_{G}(\pi^{*}\vec{f})(e)=\pi^{*}(D^{\prime}\vec{f})(e)$. Using the induction hypothesis we find $\vec{u}^{\prime}\in(\mathscr{U}\mathfrak{g})^{\mathbin{\otimes}k}$ such that $\vec{u}^{\prime\mathrm{left},(1,0)}(\pi^{*}\vec{f})(e)=\pi^{*}(D^{\prime}\vec{f})(e)$. Now

[TABLE]

proving the claim.

Assume that $D$ is in addition left-invariant. Writing $\mathrm{L}_{g}\colon G/H\to G/H$ also for the action of $g\in G$ on $G/H$ we compute

[TABLE]

Thus $\vec{u}^{\mathrm{left},(1,0)}(\pi^{*}\vec{f})=\pi^{*}(D\vec{f})$ holds for all $\vec{f}\in\mathscr{C}^{\infty}(G/H)^{k}$. Finally, we need to show that $\vec{u}$ has the correct invariance properties under the adjoint action of $H$. Define $\mathrm{R}_{g}\colon G\to G$, $\mathrm{R}_{g^{\prime}}(g)\coloneqq gg^{\prime}$. Since $\mathrm{R}_{h}^{*}\pi^{*}(D\vec{f})=\pi^{*}(D\vec{f})$ for all $h\in H$ we obtain $\mathrm{R}_{h}^{*}\vec{u}^{\mathrm{left},(1,0)}\pi^{*}\vec{f}=\vec{u}^{\mathrm{left},(1,0)}\pi^{*}\vec{f}$ and therefore

[TABLE]

for all $\vec{f}\in\mathscr{C}^{\infty}(G/H)^{k}$ and all $g\in G$, where the first equality follows as in the proof of Lemma 1.1. This means that $(\operatorname{\mathrm{Ad}}_{h}\vec{u}-\vec{u})^{\mathrm{left},(1,0)}(\pi^{*}\vec{f})=0$ for all $\vec{f}\in\mathscr{C}^{\infty}(G/H)^{k}$, and therefore the proof of injectivity implies $\operatorname{\mathrm{Ad}}_{h}\vec{u}-\vec{u}\in I$, or in other words $\vec{u}\in U_{\mathrm{inv}}$. $\boxempty$

A.2 $G$-finite functions

In this subsection we introduce $G$-finite functions on a Lie group $G$ and use them to prove Subsection 2.3. The definition of $G$-finite functions uses only abstract properties of the Lie group $G$, and is therefore independent of whether $G$ is explicitly realized by matrices or not. For complex semisimple connected Lie groups a function is $G$-finite if and only if it is a polynomial, and therefore $G$-finite functions give a characterization of polynomials that is independent of the representation.

Definition \thedefinition ($G$-finite functions).

*Let $M$ be a manifold with an action of a Lie group $G$. Then $f\in\mathscr{C}^{\infty}(M)$ is said to be $G$-finite if the vector space $\operatorname{\mathrm{span}}\{g\mathbin{\triangleright}f\mid g\in G\}$ is finite dimensional. We denote the space of $G$-finite functions on $M$ by $\mathrm{Fin}^{G}(M)$ or just by $\mathrm{Fin}(M)$ if $G$ is clear from the context. *

Here $g\mathbin{\triangleright}f$ denotes the smooth function on $M$ defined by $(g\mathbin{\triangleright}f)(m)=f(g^{-1}\mathbin{\triangleright}m)$. Below, we use this definition only for $M=G$ and the action $\mathrm{L}$ or for $M=\hat{\mathcal{O}}_{\lambda}$ and the coadjoint action, and will therefore not mention these actions explicitly.

Lemma A.4**.**

*Let $G$ be a real or complex matrix Lie group and let $\mathcal{O}_{\lambda}$ be a coadjoint orbit of $G$. Then polynomials on $G$ are $G$-finite, and polynomials on $\mathcal{O}_{\lambda}$ are also $G$-finite. *

Proof**:**

Let $P_{ij}\colon G\to\mathbbm{C}$, $X\mapsto X_{ij}$, and call such polynomials elementary in this proof. We compute $(g\mathbin{\triangleright}P_{ij})(h)=P_{ij}(g^{-1}h)=\sum_{k}(g^{-1})_{ik}h_{kj}=\sum_{k}(g^{-1})_{ik}P_{kj}(h)$ for $g\in G$, so $g\mathbin{\triangleright}P_{ij}$ is a linear combination of some elementary polynomials. If $p=P_{i_{1}j_{1}}\dots P_{i_{n}j_{n}}\in\operatorname{\mathrm{Pol}}(G)$ is a product of $n$ elementary polynomials, then $g\mathbin{\triangleright}p$ is in the linear span of products of $n$ many elementary polynomials, which is a finite dimensional space. The statement for arbitrary polynomials follows by taking linear combinations.

The action of $G$ on $\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$ is obtained by restricting the adjoint action of $G$ on $\mathrm{S}\mathfrak{g}\cong\operatorname{\mathrm{Pol}}(\mathfrak{g}^{*})$. The adjoint action preserves the degree of a symmetric tensor, so $\operatorname{\mathrm{span}}\{\operatorname{\mathrm{Ad}}_{g}X\mid g\in G\}$ is finite dimensional for any $X\in\mathrm{S}\mathfrak{g}$, and therefore $\operatorname{\mathrm{span}}\{g\mathbin{\triangleright}p\mid g\in G\}$ is finite dimensional for any $p\in\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$. $\boxempty$

Proposition \theproposition.

*Let $G$ be a complex semisimple connected Lie group with coadjoint orbit $\mathcal{O}_{\lambda}$. Then $G$-finite holomorphic functions on $\mathcal{O}_{\lambda}$ are polynomials. *

Proof**:**

$\mathrm{Hol}(\mathcal{O}_{\lambda})$* is isomorphic to $\mathrm{Hol}(G)^{G_{\lambda}}$ as a $G$-module. The restriction to a maximal compact Lie subgroup $K\subseteq G$ is an injective $K$-module homomorphism to $\mathrm{L}^{2}(K)$, the square-integrable functions on $K$ with respect to the left-invariant Haar measure, so that we may view $\mathrm{Hol}(\mathcal{O}_{\lambda})$ as a $K$-submodule of $\mathrm{L}^{2}(K)$. In particular, it is completely reducible as a $K$-module and therefore also as a $G$-module. Each irreducible module of highest weight $\nu$ appears only finitely many times in $\mathrm{L}^{2}(K)$ and thus also in $\mathrm{Hol}(\mathcal{O}_{\lambda})$.*

The scalar product of $\mathrm{L}^{2}(K)$ is $K$-invariant and therefore any irreducible modules of different highest weights are orthogonal. Restricting the scalar product to $\mathrm{Hol}(\mathcal{O}_{\lambda})$ gives that $\mathrm{Hol}(\mathcal{O}_{\lambda})^{\nu}$ is orthogonal to $\mathrm{Hol}(\mathcal{O}_{\lambda})^{\nu^{\prime}}$ if $\nu\neq\nu^{\prime}$.

Assume $f\in\mathrm{Fin}(\mathcal{O}_{\lambda})$ is holomorphic and not in $\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$. We can without loss of generality assume that $f\in\mathrm{Fin}(\mathcal{O}_{\lambda})^{\nu}$ for some weight $\nu$. (Indeed, we can write $f=\smash{\sum_{\mu}f^{\mu}}$ with $f^{\mu}\in\mathrm{Fin}(\mathcal{O}_{\lambda})^{\mu}$ and only finitely many $f^{\mu}$ are non-zero because $f$ is $G$-finite. One of these $f^{\mu}$ is not in $\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$.) We can choose $f$ orthogonal to $\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})^{\nu}$ (which is finite dimensional) and therefore orthogonal to $\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$. However, this space is dense in $\mathrm{Hol}(\mathcal{O}_{\lambda})$ because polynomials on $K$ are dense in $\mathrm{L}^{2}(K)$. So $f=0$, a contradiction. $\boxempty$

Corollary \thecorollary.

*Let $G$ be a complex semisimple connected Lie group. Then the pullback map $\pi^{*}\colon\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})\to\operatorname{\mathrm{Pol}}(G)^{G_{\lambda}}$ is an isomorphism. *

Proof**:**

We have seen in the proof of Subsection 2.3 that $\pi^{*}$ is well-defined and injective, so it only remains to show that $\pi^{*}$ is surjective. Any element $f\in\operatorname{\mathrm{Pol}}(G)^{G_{\lambda}}$ is $G$-finite by Lemma A.4. Then its image under the $G$-equivariant isomorphism $\pi_{*}\colon\mathrm{Hol}(G)^{G_{\lambda}}\to\mathrm{Hol}(\mathcal{O}_{\lambda})$ is also $G$-finite because finite dimensionality of $\operatorname{\mathrm{span}}\{g\mathbin{\triangleright}f\mid g\in G\}$ implies finite dimensionality of $\operatorname{\mathrm{span}}\{g\mathbin{\triangleright}\pi_{*}f\mid g\in G\}=\operatorname{\mathrm{span}}\{\pi_{*}(g\mathbin{\triangleright}f)\mid g\in G\}$. The previous proposition implies that the $G$-finite element $\pi_{*}f\in\operatorname{\mathrm{Pol}}(\mathcal{O}_{\lambda})$ is a polynomial. It is mapped to $f$ by $\pi^{*}$. $\boxempty$

With similar methods as in this subsection one can prove that $G$-finite functions on a complex semisimple connected Lie group $G$ coincide with polynomials on $G$. Since the definition of $G$-finite functions does not depend on a representation of $G$ as a linear group, it follows that our definition of polynomials in Subsection 2.3 is indeed independent of the representation. The same result is true for a compact semisimple connected Lie group $K$.

A.3 Complex structures on real coadjoint orbits

We have seen in Subsection 1.1 that a coadjoint orbit of a real Lie group $G$ always admits a $G$-invariant symplectic structure, in particular its dimension is even. In this subsection, we will see that a semisimple coadjoint orbit $\mathcal{O}_{\lambda}$ of a connected semisimple real Lie group $G$ admits a $G$-invariant complex structure if $G_{\lambda}$ is compact, and that the set of such complex structures is in bijection to invariant orderings. If $G$ is compact, then there is a unique $G$-invariant complex structure that makes $\mathcal{O}_{\lambda}$ a Kähler manifold. If $G$ is not compact, then $\mathcal{O}_{\lambda}$ might or might not admit a Kähler structure. All results of this subsection are classical and well-known, see for example [7] for a summary.

Let $G$ be a real connected semisimple Lie group. Assume that $\lambda\in\mathfrak{g}^{*}$ is semisimple and that $G_{\lambda}$ is compact. Then any Cartan subalgebra $\mathfrak{h}\subseteq\mathfrak{g}$ containing $\lambda^{\sharp}$ is contained in $\mathfrak{g}_{\lambda}$ and therefore compact. As usual, we denote the complexification of $\mathfrak{g}$ by $\hat{\mathfrak{g}}$ and let $\overline{{}\cdot{}\vphantom{\lambda}}$ be the complex conjugation of $\hat{\mathfrak{g}}$ with respect to $\mathfrak{g}$.

Recall that a root $\alpha\in\mathfrak{h}^{*}$ is called compact if the Killing form $B$ is negative definite on $\mathfrak{g}\cap(\mathfrak{g}^{\alpha}\oplus\mathfrak{g}^{-\alpha})$, and non-compact if it is positive definite. (The root spaces $\mathfrak{g}^{\alpha}$ are subspaces of the complexification $\hat{\mathfrak{g}}$ of $\mathfrak{g}$.) We can always choose $X_{\alpha}\in\mathfrak{g}^{\alpha}$ such that $B(X_{\alpha},X_{-\alpha})=1$ and if $[X_{\alpha},X_{\beta}]=N_{\alpha,\beta}X_{\alpha+\beta}$, then $N_{-\alpha,-\beta}=-N_{\alpha,\beta}$ (see [7, Section 3]). In this case,

[TABLE]

Recall that $\hat{\Delta}$ is the set of roots that are not orthogonal to $\lambda$.

Theorem \thethm.

*Let $\mathcal{O}_{\lambda}$ be a coadjoint orbit of a real connected semisimple Lie group $G$. Assume that $G_{\lambda}$ is compact, and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$ containing $\lambda^{\sharp}$. Then $G$-invariant complex structures on $\mathcal{O}_{\lambda}$ are in bijection with invariant orderings of $\hat{\Delta}$ (i.e. choices of positive roots $\hat{\Delta}^{+}$ that arise as $\hat{\Delta}^{+}=\hat{\Delta}\cap\Delta^{+}$ from an invariant ordering of $\Delta$ as defined in Subsection 2.2). *

Proof** (Sketch):**

Introduce $\mathfrak{m}=\bigoplus_{\alpha\in\hat{\Delta}}\mathfrak{g}^{\alpha}\cong\hat{\mathfrak{g}}/\hat{\mathfrak{g}}_{\lambda}$. Since taking fundamental vector fields (see Subsection 1.1) gives an isomorphism $\mathfrak{g}/\mathfrak{g}_{\lambda}\to\mathrm{T}_{\lambda}\mathcal{O}_{\lambda}$, $\mathfrak{m}$ is isomorphic to the complexified tangent space $\mathrm{T}_{\lambda}^{\mathbbm{C}}\mathcal{O}_{\lambda}$ and $\mathfrak{g}\cap\mathfrak{m}$ is isomorphic to $\mathrm{T}_{\lambda}\mathcal{O}_{\lambda}$.

Given an invariant ordering of $\hat{\Delta}$, see Subsection 2.2, define $I\colon\mathfrak{m}\to\mathfrak{m}$ by extending $X_{\alpha}\mapsto\mathrm{i}X_{\alpha}$ if $\alpha\in\hat{\Delta}^{+}$, $X_{\alpha}\mapsto-\mathrm{i}X_{\alpha}$ if $\alpha\in\hat{\Delta}^{-}$ linearly. Clearly $I^{2}=-\mathsf{id}$. For both a compact and a non-compact root $\alpha$, $I$ restricts to an endomorphism of $\mathfrak{g}\cap(\mathfrak{g}^{\alpha}\oplus\mathfrak{g}^{-\alpha})$, from which it follows that $I$ restricts to a map $\mathfrak{g}\cap\mathfrak{m}\to\mathfrak{g}\cap\mathfrak{m}$, squaring to $-\mathsf{id}$. To prove that it extends to a $G$-invariant almost complex structure on $\mathcal{O}_{\lambda}$, it suffices to prove that $I$ is $G_{\lambda}$-invariant. By applying the analogue of Subsection 1.1 for compact connected semisimple Lie groups to a maximally compact subgroup of $G$ containing $G_{\lambda}$, it follows that $G_{\lambda}$ is connected, and it suffices to prove that $I$ is $\mathfrak{g}_{\lambda}$-invariant, in the sense that $I([A,B])=[A,I(B)]$ holds for all $A\in\mathfrak{g}_{\lambda}$ and $B\in\mathfrak{m}$. This identity holds for $A\in\mathfrak{h}$ since $I$ preserves the root spaces. So we only need to check it for $A=X_{\alpha}$ and $B=X_{\beta}$ with $\alpha\in\Delta^{\prime}$ and $\beta\in\hat{\Delta}$, which is equivalent to the invariance of the ordering. Finally, one uses that $\alpha+\beta$ is positive if $\alpha,\beta\in\hat{\Delta}$ are positive to compute that the Nijenhuis torsion of $I$ vanishes, so $I$ is indeed a complex structure.

Vice versa, a $G$-invariant complex structure $I$ on $\mathcal{O}_{\lambda}$ determines a $\mathfrak{g}_{\lambda}$-invariant map $I\colon\mathfrak{m}\to\mathfrak{m}$ with $I^{2}=-\mathsf{id}$ by restricting to the tangent space at $\lambda$ and complexifying. In particular $I$ is $\mathfrak{h}$-invariant, and therefore preserves the root spaces, so $X_{\alpha}\mapsto\mathrm{i}c_{\alpha}X_{\alpha}$ with $c_{\alpha}=\pm 1$. Since $I$ preserves the real tangent space, we must have $c_{\alpha}=-c_{-\alpha}$. The Nijenhuis torsion of the complex structure vanishes, which implies that $\hat{\Delta}^{+}=\{\alpha\in\hat{\Delta}\mid c_{\alpha}=1\}$ defines an ordering. Finally invariance under the whole Lie algebra $\mathfrak{g}_{\lambda}$ gives that this ordering is invariant. $\boxempty$

Proposition \theproposition.

*If $\mathcal{O}_{\lambda}$ is a coadjoint orbit of a compact connected semisimple Lie group $K$, then $\mathcal{O}_{\lambda}$ has a unique $K$-invariant complex structure $I$ that makes $(\mathcal{O}_{\lambda},I,\omega_{\mathrm{KKS}})$ a Kähler manifold, and this complex structure corresponds to an ordering for which $\alpha\in\hat{\Delta}$ is positive if and only if $(\alpha,\mathrm{i}\lambda)>0$. *

Note that $\alpha$ attains purely imaginary values on $\mathfrak{k}$, whereas $\lambda$ attains real values, so $(\alpha,\mathrm{i}\lambda)\in\mathbbm{R}$. The ordering for which $\alpha\in\hat{\Delta}$ is positive if $(\alpha,\mathrm{i}\lambda)>0$ is standard (see Subsection 2.2).

Proof**:**

Since $K$ is compact, it follows that any root is compact. Given a $K$-invariant complex structure $I$, we associate the (not necessarily positive definite) metric $g(v,w)=\omega_{\mathrm{KKS}}(v,Iw)$ and $\mathcal{O}_{\lambda}$ is a Kähler manifold if $g$ is positive definite. Since $I$ and $\omega_{\mathrm{KKS}}$ are $K$-invariant, so is $g$ and we may check positive definiteness on $\mathrm{T}_{\lambda}\mathcal{O}_{\lambda}$. Identifying $\mathrm{T}^{\mathbbm{C}}_{\lambda}\mathcal{O}_{\lambda}$ with $\mathfrak{m}$ as in the proof of the previous proposition and extending $g$ complex linearly, we compute that $g(X_{\alpha},X_{\beta})=\omega_{\mathrm{KKS}}(X_{\alpha},IX_{\beta})=c_{\beta}\lambda([X_{\alpha},X_{\beta}])$ for all $\alpha,\beta\in\hat{\Delta}$. This expression is non-zero only if $\alpha=-\beta$, and in this case $g(X_{\alpha},X_{-\alpha})=-\mathrm{i}c_{\alpha}\lambda(\alpha^{\sharp})=-\mathrm{i}c_{\alpha}\cdot(\alpha,\lambda)$. Then

[TABLE]

So $g$ is positive definite if and only if $c_{\alpha}=1$ for all $\alpha\in\hat{\Delta}$ with $(\alpha,\mathrm{i}\lambda)>0$. $\boxempty$

Note that the situation is more complicated if $G$ is non-compact, but $G_{\lambda}$ is compact, since we may then have both compact and non-compact roots. The condition for $g$ being positive definite then becomes $c_{\alpha}=1$ if either $\alpha$ is a compact root and $(\alpha,\mathrm{i}\lambda)>0$ or if $\alpha$ is a non-compact root and $(\alpha,\mathrm{i}\lambda)<0$. If these conditions define an invariant ordering, then $\mathcal{O}_{\lambda}$ has a $G$-invariant Kähler structure (which is automatically unique). One can give more explicit criteria for when the conditions above define an invariant ordering, see [7], but we only need the following easy case.

Corollary \thecorollary.

*Let $\mathcal{O}_{\lambda}$ be a coadjoint orbit of a connected semisimple Lie group $G$. Assume that $G_{\lambda}$ is compact, and that $\mathfrak{h}$ is a Cartan subalgebra containing $\lambda^{\sharp}$. If all roots in $\hat{\Delta}$ are non-compact, then $(\mathcal{O}_{\lambda},I,\omega_{\mathrm{KKS}})$ is a Kähler manifold, where $I$ is the complex structure corresponding to the ordering for which $\alpha\in\hat{\Delta}$ is positive if and only if $(\alpha,\mathrm{i}\lambda)<0$. *

Acknowledgements

The author would like to thank Matthias Schötz for many valuable discussions and helpful comments on an earlier version of this article. He is extremely grateful to his advisor Ryszard Nest for many helpful and inspiring discussions on the content of this paper and related topics.

Funding

The author was supported by the Danish National Research Foundation through the Centre of Symmetry and Deformation (DNRF92).