Quasi-polynomials and the singular $[Q,R]=0$ theorem

TL;DR

This paper revisits the `shift-desingularization' version of the $[Q,R]=0$ theorem for singular symplectic quotients, using quasi-polynomial behavior and index formulas to analyze multiplicities in geometric quantization.

Contribution

It introduces an approach combining the Szenes-Vergne proof with Berline-Vergne index formula to analyze multiplicities for singular quotients, extending previous methods.

Findings

Demonstrates quasi-polynomial behavior of multiplicities

Adapts stationary phase expansion for singular cases

Provides a new perspective on the $[Q,R]=0$ theorem

Abstract

In this short note we revisit the `shift-desingularization' version of the $[Q,R]=0$ theorem for possibly singular symplectic quotients. We take as starting point an elegant proof due to Szenes-Vergne of the quasi-polynomial behavior of the multiplicity as a function of the tensor power of the prequantum line bundle. We use the Berline-Vergne index formula and the stationary phase expansion to compute the quasi-polynomial, adapting an early approach of Meinrenken.

Equations

$${\mu }_{\mathfrak{g}}:M\to {\mathfrak{g}}^{\ast },$$

$$\iota (X_M)\omega =-\mathrm{d}⟨{\mu }_{\mathfrak{g}},X⟩,X\in \mathfrak{g}.$$

$${\mathcal{L}}_X^L-{\nabla }_{X_M}^L=2\pi \mathrm{i}⟨{\mu }_{\mathfrak{g}},X⟩.$$

$${\mu }_{\mathfrak{g}}{|}_Y:Y\to I$$

$${\text{index}}_G(D_{L^k})=\sum _{\lambda \in \Lambda \cap {\mathfrak{t}}_{+}^{\ast }}{m}_G(k,\lambda ){\chi }_{\lambda }.$$

$$P={\mu }_{\mathfrak{g}}^{-1}(\xi )$$

$${\Sigma }_g=P^g/(T/T_I),\Sigma =\underset{g\in S_P}{⨆}{\Sigma }_g.$$

$$\mathcal{D}_{\mathbb{C}}^{g}(\nu_{\Sigma_{g},\Sigma})=\textnormal{det}_{\mathbb{C}}\big{(}1-g_{\nu}^{-1}{\rm e}^{-\frac{{\mathrm{i}}}{2\pi}F_{\nu}}\big{)},$$

$$L_{{\Sigma }_g}=(L{|}_{P^g})/(T/T_I),L_{\Sigma }=\underset{g\in S_P}{⨆}{L}_{{\Sigma }_g}.$$

$$g_L:{\Sigma }_g\to U(1)$$

$$C_{\mathfrak{p}}=\{(t,t\tau )|t\in (0,\infty ),\tau \in \mathfrak{p}\}\subset \mathbb{R}\times {\mathfrak{t}}^{\ast }.$$

$$m_G(k,\lambda )=\sum _{g\in S_P}{g}^{-\lambda }{\int }_{{\Sigma }_g}\frac{1}{d}\frac{g_L^k\text{Ch}(L_{\Sigma }{)}^k\text{Td}(\Sigma )}{{\mathcal{D}}_{\mathbb{C}}^g({\nu }_{{\Sigma }_g,\Sigma })}{\mathrm{e}}^{⟨\lambda ,F_{\theta }⟩}.$$

$$(\mathfrak{g}/\mathfrak{t}{)}^{1,0}\simeq \underset{\alpha \in {\mathcal{R}}_{+}}{⨁}{\mathfrak{g}}_{\alpha }.$$

$$\displaystyle\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}/\mathfrak{t}}\big{(}1-t^{-1}\big{)}=\prod_{\alpha\in\mathcal{R}_{+}}\big{(}1-t^{-\alpha}\big{)}=\prod_{\alpha\in\mathcal{R}_{+}}\big{(}1-{\rm e}^{-2\pi{\mathrm{i}}\langle\alpha,X\rangle}\big{)},$$

$${\text{det}}_{\mathbb{C}}^{\mathfrak{g}/\mathfrak{t}}(-X)=\prod _{\alpha \in {\mathcal{R}}_{+}}-2\pi \mathrm{i}⟨\alpha ,X⟩.$$

$$\displaystyle\chi_{\lambda}(t)\cdot\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}/\mathfrak{t}}\big{(}1-t^{-1}\big{)}=\sum_{w\in W}(-1)^{l(w)}t^{w(\lambda+\rho)-\rho},$$

$$m_{\lambda }(w(\mu +\rho )-\rho )=(-1{)}^{l(w)}{m}_{\lambda }(\mu ).$$

$$Q(k,t)=\sum _{\lambda \in \Lambda }m(k,\lambda )t^{\lambda }.$$

$$\displaystyle Q(k,t)=\sum_{\lambda\in\Lambda}m(k,\lambda)t^{\lambda}=\textnormal{index}_{T}(D_{L^{k}})(t)\cdot\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}/\mathfrak{t}}\big{(}1-t^{-1}\big{)}.$$

$$\mu ={\text{pr}}_{{\mathfrak{t}}^{\ast }}\circ {\mu }_{\mathfrak{g}}$$

$$\displaystyle Q(k,t)=\sum_{F\subset M^{T}}t^{k\mu_{F}}\int_{F}\frac{{\rm e}^{k\omega}\textnormal{Td}(F)}{\mathcal{D}_{\mathbb{C}}^{t}(\nu_{F})}\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}/\mathfrak{t}}\big{(}1-t^{-1}\big{)},$$

$$m=\sum _C{m}_C,$$

$${\mathbf{A}}_C=\{(t,t\tau )|t\in {\mathbb{R}}_{>0},\tau \in A_C\}\subset \mathbb{R}\times {\mathfrak{t}}^{\ast }.$$

$$⟨{\tau }_C,\lambda ⟩\ge k⟨{\tau }_C,{\gamma }_C⟩+⟨{\tau }_C,{\sigma }_C⟩,{\sigma }_C:=\sum _{\begin{array}{c}\delta \in \text{wt}({\nu }_C)\\ ⟨{\tau }_C,\delta ⟩>0\end{array}}\delta -\sum _{\begin{array}{c}\alpha \in {\mathcal{R}}_{+}\\ ⟨{\tau }_C,\alpha ⟩>0\end{array}}\alpha .$$

$$C_{\mathfrak{p}}=\{(t,t\tau )|t\in (0,\infty ),\tau \in \mathfrak{p}\}\subset \mathbb{R}\times {\mathfrak{t}}^{\ast }.$$

$$H_C=\{\xi |⟨{\tau }_C,\xi ⟩\le ⟨{\tau }_C,{\gamma }_C⟩\}.$$

$${\sigma }_C=\sum _{\begin{array}{c}\delta \in \text{wt}({\nu }_C)-{\mathcal{R}}_{+}^{{\tau }_C}\\ ⟨{\tau }_C,\delta ⟩>0\end{array}}\delta ,$$

$$⟨{\tau }_C,{\sigma }_C⟩\ge 0$$

$$⟨{\tau }_C,\alpha ⟩=⟨{\gamma }_C,\alpha ⟩-⟨\gamma ,\alpha ⟩>0.$$

$$⟨{\tau }_C,{\gamma }_I⟩<⟨{\tau }_C,{\gamma }_C⟩$$

$$U=G_{\sigma }\cdot \bigcup _{\overline{\tau }\supset \sigma }\tau ,$$

Full text

\FirstPageHeading

\ShortArticleName

Quasi-Polynomials and the Singular $[Q,R]=0$ Theorem

\ArticleName

Quasi-Polynomials and the Singular $\bm{[Q,R]=0}$

Theorem

\Author

Yiannis LOIZIDES

\AuthorNameForHeading

Y. Loizides

\Address

Pennsylvania State University, USA \Emailyxl649@psu.edu

\ArticleDates

Received July 16, 2019, in final form November 13, 2019; Published online November 18, 2019

\Abstract

In this short note we revisit the ‘shift-desingularization’ version of the $[Q,R]=0$ theorem for possibly singular symplectic quotients. We take as starting point an elegant proof due to Szenes–Vergne of the quasi-polynomial behavior of the multiplicity as a function of the tensor power of the prequantum line bundle. We use the Berline–Vergne index formula and the stationary phase expansion to compute the quasi-polynomial, adapting an early approach of Meinrenken.

\Keywords

symplectic geometry; Hamiltonian $G$-spaces; symplectic reduction; geometric quantization; quasi-polynomials; stationary phase

\Classification

53D20; 53D50

1 Introduction

Let $(M,\omega)$ be a compact connected symplectic manifold equipped with an action of a compact connected Lie group $G$ by symplectomorphisms. Suppose that the action of $G$ is Hamiltonian, meaning that there is a $G$-equivariant map, the moment map,

[TABLE]

where $\mathfrak{g}^{\ast}$ is the dual of the Lie algebra $\mathfrak{g}=\textnormal{Lie}(G)$, satisfying the moment map condition

[TABLE]

Let \big{(}L,\nabla^{L}\big{)} be a $G$-equivariant prequantum line bundle with connection on $M$, i.e., $L$ is a $G$-equivariant Hermitian line bundle with compatible connection $\nabla^{L}$, \big{(}\nabla^{L}\big{)}^{2}=-2\pi{\mathrm{i}}\omega and the derivative of the $G$-action on $L$ satisfies Kostant’s condition

[TABLE]

Choose a compatible almost complex structure $J$ on $M$, i.e., $\omega(Jw,Jv)=\omega(w,v)$ and $\omega(w,Jv)\allowbreak=:g(w,v)$ is a Riemannian metric. Let $D_{L}$ denote the Dolbeault–Dirac operator twisted by \big{(}L,\nabla^{L}\big{)}, an elliptic differential operator acting on sections of the spinor bundle $\wedge T^{\ast}_{0,1}M\otimes L$. The kernel of $D_{L}$ carries an action of $G$, and the $G$-equivariant index is defined to be the difference \textnormal{index}_{G}(D_{L}):=\textnormal{ker}\big{(}D_{L}^{\textnormal{even}}\big{)}-\textnormal{ker}\big{(}D_{L}^{\textnormal{odd}}\big{)} of the kernel of $D_{L}$ on even/odd degree forms, regarded as an element of the representation ring $R(G)$.

The quantization-commutes-with-reduction theorem ($[Q,R]=0$ theorem) describes the multiplicity of the trivial representation in $\textnormal{index}_{G}(D_{L})$ in terms of the symplectic quotient $M^{\textnormal{red}}:=\mu_{\mathfrak{g}}^{-1}(0)/G$. When [math] is a regular value of $\mu_{\mathfrak{g}}$, $M^{\textnormal{red}}$ is an orbifold and the theorem states that $\textnormal{index}_{G}(D)^{G}$ equals the index of the twisted Dolbeault–Dirac operator $D^{\textnormal{red}}_{L^{\textnormal{red}}}$ on $M^{\textnormal{red}}$. The theorem was first conjectured by Guillemin–Sternberg [3], and the general case ($M$, $G$ both compact, [math] a regular value) was first proved by Meinrenken [8]. Different proofs of the $[Q,R]=0$ theorem were given by Tian–Zhang [15] and Paradan [11]. The theorem has since been extended in various directions.

There are versions of the $[Q,R]=0$ theorem when [math] is not necessarily a regular value, due to Meinrenken–Sjamaar [10]; below we will give a precise statement of one of these results, involving a partial shift desingularization, i.e., $\textnormal{index}_{G}(D_{L})^{G}$ is related to the index on the symplectic quotient at a nearby weakly regular value. At the same time, we introduce some notation that will be of use later on.

Fix a maximal torus $T$ with Lie algebra $\mathfrak{t}$. Let $\Lambda\subset\mathfrak{t}^{\ast}$ be the (real) weight lattice. Given $\lambda\in\Lambda$, the corresponding character $T\rightarrow U(1)$ is written $t\mapsto t^{\lambda}={\rm e}^{2\pi{\mathrm{i}}\langle\lambda,X\rangle}$ where $t={\rm e}^{X}$, $X\in\mathfrak{t}$. Let $\mathcal{R}\subset\Lambda$ be the set of roots. We also fix a closed positive Weyl chamber $\mathfrak{t}_{+}$, which determines a set of positive (resp. negative) roots $\mathcal{R}_{\pm}$. For each relatively open face $\sigma\subset\mathfrak{t}_{+}^{\ast}$, the stabilizer $G_{\xi}$ of points $\xi\in\sigma$ under the coadjoint action, does not depend on $\xi$, and will be denoted $G_{\sigma}$. If $\sigma_{1}\subset\overline{\sigma}_{2}$ then $G_{\sigma_{1}}\supset G_{\sigma_{2}}$. Note also that $G_{\sigma}$ is connected and contains the maximal torus $T$. The Lie algebra $\mathfrak{g}_{\sigma}$ decomposes into its semi-simple and central parts $\mathfrak{g}_{\sigma}=[\mathfrak{g}_{\sigma},\mathfrak{g}_{\sigma}]\oplus\mathfrak{z}_{\sigma}$. The subspace $\mathfrak{z}_{\sigma}^{\ast}\subset\mathfrak{t}^{\ast}$ is defined to be the annihilator of $[\mathfrak{g}_{\sigma},\mathfrak{g}_{\sigma}]$, or equivalently the fixed point set of the coadjoint $G_{\sigma}$ action. The face $\sigma$ is an open subset of $\mathfrak{z}_{\sigma}^{\ast}$.

Let $\Delta=\mu_{\mathfrak{g}}(M)\cap\mathfrak{t}_{+}^{\ast}$ be the moment polytope. A well-known theorem in symplectic geometry states that there is a unique face $\sigma\subset\mathfrak{t}^{\ast}_{+}$ of minimal dimension such that $\Delta\subset\overline{\sigma}$ (briefly, this is a consequence of (1.1), which implies that ${\rm d}\mu_{\mathfrak{g}}$ has constant rank on the top dimensional $G$-orbit type stratum, and the complement of the latter has codimension at least $2$); $\sigma$ is called the principal face or principal wall. The corresponding symplectic cross-section, called the principal cross-section, $Y=\mu_{\mathfrak{g}}^{-1}(\sigma)$ is a Hamiltonian $G_{\sigma}$-space. Moreover the semi-simple part $[G_{\sigma},G_{\sigma}]$ of $G_{\sigma}$ acts trivially on $Y$. For further details, see for example [5] and references therein.

Let $I\subset\mathfrak{z}_{\sigma}^{\ast}$ be the smallest affine subspace containing $\Delta$. Let $\mathfrak{t}_{I}\subset\mathfrak{t}$ be the annihilator of the subspace parallel to $I$, and let $T_{I}=\exp(\mathfrak{t}_{I})\subset T$ be the corresponding subtorus. By equation (1.1), $\mathfrak{t}_{I}$ is the generic infinitesimal stabilizer of $Y$. In particular $T_{I}$ acts trivially, hence the quotient torus $T/T_{I}$ acts on $Y$. The moment map $\mu_{\mathfrak{g}}$ may have no non-trivial regular values. But the restriction

[TABLE]

viewed as a map with codomain $I$, always has non-trivial regular values, and we will refer to these as weakly-regular values. If $\xi$ is a weakly-regular value, then the reduced space $M_{\xi}=\mu_{\mathfrak{g}}^{-1}(\xi)/G_{\sigma}$ is an orbifold. Let $L_{\xi}=L|_{\mu_{\mathfrak{g}}^{-1}(\xi)}/G_{\sigma}$ be the corresponding (orbifold) line bundle over $M_{\xi}$.

Theorem 1.1** ([10], see also [11, 13]).**

Let $(M,\omega,\mu_{\mathfrak{g}})$ be a compact connected Hamiltonian $G$-space with moment polytope $\Delta$. If $0\notin\Delta$ then $\textnormal{index}_{G}(D_{L})^{G}=0$. Otherwise for every weakly-regular value $\xi\in\Delta$ sufficiently close to [math], $\textnormal{index}_{G}(D_{L})^{G}$ equals the index of the Dolbeault–Dirac operator $D^{\textnormal{red}}_{L_{\xi}}$ on the reduced space $M_{\xi}$.

We will now describe the main result of this article and its relation to Theorem 1.1. Consider tensor powers $L^{k}$, $k\in\mathbb{Z}_{>0}$ of the prequantum line bundle. For a dominant weight $\lambda$, let $\chi_{\lambda}\in R(G)$ denote the character of the irreducible representation of $G$ with highest weight $\lambda$. We define the multiplicity function $m_{G}(k,\lambda)$ by the expression

[TABLE]

An important theme in the work of Szenes–Vergne [14] and also in our approach, is that the function $m_{G}(k,\lambda)$ has more coherent behavior than its restriction to any fixed value of $k$.

The statement of the result requires some further background on orbifolds, for which we refer the reader to, for example, [2, Appendix A], [8, Section 2]. A small warning is that we will not require the action of isotropy groups in orbifold charts to be effective (this is in agreement with the references [2, 8] mentioned above). One advantage of permitting this, is that for a locally free action of a compact Lie group $K$ on a manifold $P$, the corresponding orbifold $P/K$ has orbifold charts given automatically by the slice theorem, with the isotropy groups being simply the isotropy groups for the action of $K$ on $P$.

In fact all the orbifolds that we will encounter arise naturally as such quotients $P/K$, and one could avoid mentioning orbifolds altogether by working instead with suitable $K$-basic structures on $P$. An example is the description of characteristic forms for orbifold vector bundles, which can be defined in terms of orbifold charts for $P/K$, or alternatively in terms of $K$-basic differential forms on $P$. In brief, the latter approach goes as follows. One can take the complex $(\Omega_{\textnormal{bas}}(P),{\rm d})$ of $K$-basic differential forms on $P$ as a working definition of the de Rham complex of $P/K$ (if $K$ acts freely then $P/K$ is a manifold and pullback of forms from $P/K$ to $P$ is an isomorphism of complexes $(\Omega(P/K),{\rm d})\simeq(\Omega_{\textnormal{bas}}(P),{\rm d})$). A $K$-equivariant vector bundle $E\rightarrow P$ determines an orbifold vector bundle $E/K$ over $P/K$. Let $\theta$ be a connection on $P$ with curvature $F_{\theta}$. The choice of connection determines a Cartan map (cf. [9]) from closed $K$-equivariant forms $\alpha(X)$ on $P$ to closed $K$-basic forms: $\alpha(X)\mapsto\textnormal{Car}_{\theta}(\alpha):=\Pi_{\textnormal{hor}}\alpha(F_{\theta})$, where $\Pi_{\textnormal{hor}}$ is the projection onto the horizontal part relative to the connection. The Cartan map induces an isomorphism from the $K$-equivariant cohomology of $P$ to the cohomology of the complex of basic differential forms on $P$. If $\alpha(X)$ is a $K$-equivariant characteristic form (constructed via the $K$-equivariant analogue of the usual Chern–Weil construction cf. [1, 9]), then one may take $\textnormal{Car}_{\theta}(\alpha)\in\Omega_{\textnormal{bas}}(P)$ as the definition of the corresponding characteristic form for $E/K$.

Let $\xi\in\Delta$ be a weakly-regular value. By the moment map equation (1.1), the action of $K=T/T_{I}$ on the level set

[TABLE]

is locally free. The set $S_{P}$ of elements $g\in T/T_{I}$ such that $P^{g}\neq\varnothing$ is finite. For each $g\in S_{P}$, we obtain an orbifold

[TABLE]

Note that $\Sigma_{1}=P/(T/T_{I})=M_{\xi}$ identifies with the reduced space itself, and more generally $\Sigma_{g}$ identifies with a symplectic quotient of $Y^{g}$. For each $g\in S_{P}$ there is an immersion $\Sigma_{g}\hookrightarrow\Sigma$ induced by $P^{g}\hookrightarrow P$. Let $\nu_{\Sigma_{g},\Sigma}$ denote the (orbifold) normal bundle (the quotient $\nu_{P^{g},P}/(T/T_{I})$), which inherits a complex structure from the almost complex structures on $Y$, $Y^{g}$. Define the characteristic form

[TABLE]

where $g_{\nu}$ denotes the action of $g$ on the normal bundle (defined in terms of an orbifold chart, or in terms of $\nu_{P^{g},P}$), and $F_{\nu}$ denotes the curvature. Taking the quotient of $L|_{P^{g}}$ we obtain (orbifold) line bundles

[TABLE]

There is a locally constant function

[TABLE]

giving the phase of the action of $g$ on $L_{\Sigma_{g}}$ (or equivalently on $L|_{P^{g}}$). Let $d\colon\Sigma\rightarrow\mathbb{Z}$ be the locally constant function giving the size of a generic isotropy group for $\Sigma$ (or equivalently the number of elements in the generic stabilizer for the $T/T_{I}$ action on $\sqcup P^{g}$).

Let $\theta$ be a connection for the locally free $K=T/T_{I}$-action on $\sqcup_{g\in S_{P}}P^{g}$. The curvature $F_{\theta}$ is horizontal and $\mathfrak{t}/\mathfrak{t}_{I}$-valued, hence for any $\lambda\in(\mathfrak{t}/\mathfrak{t}_{I})^{\ast}=I$, the form $\langle\lambda,F_{\theta}\rangle$ is $K$-basic, hence descends to $\Sigma$. With the preparations above, we can state the main result of this note.

Theorem 1.2**.**

If $0\notin\Delta$ then $m(k,0)=0$ for all $k\geq 1$. If $0\in\Delta$ then there is a closed polytope $\mathfrak{p}\subset\Delta$ of the same dimension as $\Delta$ and containing the origin such that the following is true. Let $C_{\mathfrak{p}}$ denote the cone

[TABLE]

Fix a weakly regular value $\xi\in\Delta$ sufficiently close to [math] as in Theorem 1.1. Let $P=\mu_{\mathfrak{g}}^{-1}(\xi)$ and define $\Sigma$, $L_{\Sigma}$, etc. as above. Then for all $(k,\lambda)\in(\mathbb{Z}_{>0}\times\Lambda)\cap C_{\mathfrak{p}}$,

[TABLE]

Of course this result is also originally due to Meinrenken–Sjamaar [10]. Theorem 1.1 follows immediately from Theorem 1.2 by applying Kawasaki’s index theorem for orbifolds to \textnormal{index}\big{(}D^{\textnormal{red}}_{L_{\xi}}\big{)} and comparing with the evaluation of (1.3) at $(k,\lambda)=(1,0)$.

Let us give a brief summary of our approach to deriving Theorem 1.2. Recall that a function $f$ on a lattice $\Gamma$ in a real vector space $V$ is said to be quasi-polynomial if there is a sublattice $\Gamma^{\prime}$ with $\Gamma/\Gamma^{\prime}$ finite and $f$ restricts to a polynomial function on each coset of $\Gamma^{\prime}$. More generally, one says $f$ is quasi-polynomial on a subset $\Gamma_{0}\subset\Gamma$ if $f\upharpoonright\Gamma_{0}=q\upharpoonright\Gamma_{0}$ for some quasi-polynomial $q$. A fundamental fact, originally derived from Theorem 1.1 by Meinrenken–Sjamaar [10], is that $m_{G}$ is quasi-polynomial on the subset $C_{\mathfrak{p}}\cap(\mathbb{Z}_{>0}\times\Lambda)$. Our first goal, in Section 2, is to give an independent proof of this fact, taking as a starting point a formula for $m_{G}$ due to Szenes–Vergne [14] (inspired by work of Paradan [11]), which they obtained by a combinatorial rearrangement of the fixed-point formula for the index.

Then in Section 3 we adapt an idea of Meinrenken [7] to compute the quasi-polynomial $m_{G}\upharpoonright C_{\mathfrak{p}}$ using the Berline–Vergne index formula and the principle of stationary phase. The output of the stationary phase formula is an asymptotic expansion for $m_{G}(k,k\xi)$ in powers of $k$ (allowing coefficients that are periodic in $k$). As one knows in advance that $m_{G}(k,k\xi)$ is quasi-polynomial in $k$, one concludes that the expansion is exact, yielding Theorem 1.2.

The article of Meinrenken–Sjamaar [10] contains, besides Theorem 1.1, a wealth of detailed information about singular reduction and $[Q,R]=0$. Our goal in this short note is much more modest. We also do not make a great claim of originality, and in particular the debt to [14] and [7] will be apparent. Part of our motivation stems from the hope that the article of Szenes–Vergne [14], in combination with this note, will provide a more elementary treatment of the $[Q,R]=0$ theorem than was previously available.

2 Quasi-polynomials and the multiplicity function

The goal of this section is Theorem 2.2 on the quasi-polynomial behavior of the multiplicity function, which we prove using results of Szenes–Vergne [14] reviewed below.

The quotient $\mathfrak{g}/\mathfrak{t}$ can be identified with the unique $\textnormal{Ad}(T)$-invariant complement to $\mathfrak{t}$ in $\mathfrak{g}$. Let $\mathfrak{t}\subset\mathfrak{h}\subset\mathfrak{g}$ be a $T$-invariant subspace. We may similarly identify $\mathfrak{h}/\mathfrak{t}$ and $\mathfrak{g}/\mathfrak{h}$ with subspaces of $\mathfrak{g}$. The choice of positive roots $\mathcal{R}_{+}$ determines a complex structure on $\mathfrak{g}/\mathfrak{t}$, whose $+{\mathrm{i}}$-eigenspace is identified with the direct sum of the positive root spaces:

[TABLE]

We obtain similar complex structures on $\mathfrak{g}/\mathfrak{h}$, $\mathfrak{h}/\mathfrak{t}$, whose $+{\mathrm{i}}$-eigenspaces are direct sums of positive roots spaces. We will write $\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}/\mathfrak{t}}(a)$ (resp. $\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}/\mathfrak{h}}(a)$, $\textnormal{det}_{\mathbb{C}}^{\mathfrak{h}/\mathfrak{t}}(a)$) for the determinant of a complex linear endomorphism $a$ of $\mathfrak{g}/\mathfrak{t}$ (resp. $\mathfrak{g}/\mathfrak{h}$, $\mathfrak{h}/\mathfrak{t}$). An example is the endomorphism $\textnormal{Ad}_{t}$, $t\in T$ (resp. $\textnormal{ad}_{X}$, $X\in\mathfrak{t}$); in this case we will simply write $\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}/\mathfrak{t}}(t)$ instead of $\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}/\mathfrak{t}}(\textnormal{Ad}_{t})$ (resp. $\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}/\mathfrak{t}}(X)$ instead of $\textnormal{det}_{\mathbb{C}}^{\mathfrak{g}/\mathfrak{t}}(\textnormal{ad}_{X})$), the action of $T$ (resp. $\mathfrak{t}$) on $\mathfrak{g}/\mathfrak{t}$ being understood. Then for example if $t={\rm e}^{X}\in T$,

[TABLE]

For $\lambda\in\Lambda\cap\mathfrak{t}_{+}^{\ast}$, the Weyl character formula says that for $t\in T$,

[TABLE]

where $W$ is the Weyl group, $l(w)$ is the length of the element $w\in W$, and $\rho$ is the half sum of the positive roots. The right-hand-side is an element of $R(T)$ with multiplicity function $m_{\lambda}$ obtained by Fourier transform. Note that

• •

$m_{\lambda}$ is anti-symmetric under the $\rho$-shifted action of the Weyl group:

[TABLE]

• •

The support of $m_{\lambda}|_{\Lambda\cap\mathfrak{t}_{+}^{\ast}}$ is $\{\lambda\}$, where it takes the value $1$.

Conversely these two properties determine $m_{\lambda}$: it is the unique $W$-anti-symmetric function on $\Lambda$ extending the multiplicity function of $\chi_{\lambda}$. Applying these observations to the multiplicity function $m_{G}$ defined in (1.2), we make the following definition.

Definition 2.1**.**

Let $m(k,-)\colon\Lambda\rightarrow\mathbb{Z}$ be the unique $\rho$-shifted $W$-anti-symmetric function such that $m(k,\lambda)=m_{G}(k,\lambda)$ for all $\lambda\in\Lambda\cap\mathfrak{t}_{+}^{\ast}$. The corresponding character $Q(k,-)\colon T\rightarrow\mathbb{C}$ is defined as the inverse Fourier transform:

[TABLE]

Using the Weyl character formula (2.1) and the definition of $m_{G}$, it is easy to verify that

[TABLE]

We define

[TABLE]

to be the composition of the moment map $\mu_{\mathfrak{g}}$ with the projection to $\mathfrak{t}^{\ast}$. Then $\mu$ is a moment map for the action of $T$ on $M$. Suppose $t\in T$ is sufficiently generic, so that $M^{t}=M^{T}$. The Atiyah–Bott–Segal formula for the index yields

[TABLE]

where the sum is over connected components $F$ of $M^{T}$, and $\mu_{F}$ denotes the constant value of the moment map $\mu$ on $F$. The multiplicity $m$ is obtained by Fourier transform of (2.2).

Key to the approach in [14] is a different expression for $m(k,\lambda)$ that we briefly describe here. The formula depends on the choice of an invariant inner product on $\mathfrak{g}$, as well as a generic point $\gamma$ contained in $\mathfrak{t}^{\ast}_{+}$ and sufficiently close to [math] (see [14, Section 4.1] for the meaning of ‘generic’ here). Using the inner product we identify $\mathfrak{t}\simeq\mathfrak{t}^{\ast}$. We need some additional notation:

• •

Let $\textnormal{Comp}_{T}(M)$ denote the set of connected components of $M^{H}$, as $H$ ranges over all (connected) sub-tori of $T$.

• •

For $C\in\textnormal{Comp}_{T}(M)$, let $\mathfrak{t}_{C}\subset\mathfrak{t}$ be its generic infinitesimal stabilizer. Let $A_{C}$ be the smallest affine subspace containing the image $\mu(C)$. In particular $A_{M}$ is the smallest affine subspace containing $\mu(M)$. Note that $A_{C}$ is a translate of the annihilator of $\mathfrak{t}_{C}$.

• •

Let $\gamma_{C}\in A_{C}$ be the orthogonal projection of $\gamma$ onto $A_{C}$, and let $\tau_{C}=\gamma_{C}-\gamma$.

The Szenes–Vergne–Paradan formula [14, equation (39)] (see also [14, Proposition 41, Theorem 48]) is a sum of contributions:

[TABLE]

where $C$ ranges over components $C\in\textnormal{Comp}_{T}(M)$ such that $\gamma_{C}\in\mu_{\mathfrak{g}}(C)$. Szenes–Vergne derive this formula directly from (2.2) using an interesting combinatorial rearrangement, the main ingredient of which is a decomposition formula for Kostant-type partition functions. The formula is inspired by, and closely related to, the work of Paradan [11]. The fact that only a subset of the components in $\textnormal{Comp}_{T}(M)$ contribute is non-trivial and quite important for $[Q,R]=0$. The proof given by Szenes–Vergne involves studying the asymptotic behavior of the $m_{C}$’s using the Berline–Vergne formula and the principle of stationary phase. It goes back to results of Paradan [11], who proved a closely related result using transversally elliptic symbols and K-theoretic methods. Note that Szenes–Vergne assume for simplicity that $M^{T}$ consists of isolated fixed points, but it is not difficult to handle the general case with the same methods; see for example [6, Section 7] for some indications of how this can be done.

For the proof of Theorem 2.2 we do not need the precise definition of the terms $m_{C}$ in (2.3), but we will need the following two crucial properties:

The function $m_{C}$ restricts to a quasi-polynomial on each $\Lambda$-translate of the set $(\mathbb{Z}\times\Lambda)\cap\bm{A}_{C}$, where

[TABLE] 2. 2.

Let $\textnormal{wt}(\nu_{C})$ denote the list of complex weights (for the compatible almost complex structure $J$) for the $\mathfrak{t}_{C}$ action on the normal bundle $\nu_{C}$. If $\lambda\in\Lambda$ is in the support of $m_{C}(k,-)$ then $\lambda$ satisfies the inequality

[TABLE]

See the proof of [14, Theorem 49]. Note that, except for the special case $\tau_{C}=0$, (2.4) defines a half-space in $\mathfrak{t}^{\ast}$.

We will refer to these two properties as ‘property (a)’, ‘property (b)’ in the proof of the next result. Theorem 2.2 is a strengthening of [14, Theorem 49] (which says that the function $k\mapsto m(k,0)$ is quasi-polynomial), and our arguments are based on their elegant approach.

Theorem 2.2** ([10], see also [11, 12, 13]).**

If $0\notin\Delta$ then $m(k,0)=0$ for all $k\geq 1$. If $0\in\Delta$ then there is a closed polytope $\mathfrak{p}\subset\Delta$ of the same dimension as $\Delta$ and containing the origin such that $m(k,\lambda)$ is quasi-polynomial on the set of integral points $(\mathbb{Z}\times\Lambda)\cap C_{\mathfrak{p}}$ contained in the cone

[TABLE]

Proof.

The strategy is based on choosing a suitable $\gamma\in\mathfrak{t}^{\ast}_{+}$ and then analyzing the supports of the contributions $m_{C}$ to $m$ in the corresponding Szenes–Vergne–Paradan formula (2.3) using property (b). The contribution $m_{C}$ appears in (2.3) only if $\gamma_{C}\in\mu_{\mathfrak{g}}(M)\cap\mathfrak{t}^{\ast}=W\cdot\Delta\subset W\cdot I$ (recall by definition $I$ is the smallest affine subspace containing $\Delta$). Because $\gamma$ is chosen generically, the only $C\in\textnormal{Comp}_{T}(M)$ which may contribute to (2.3) are those such that the affine subspace $A_{C}$ is entirely contained in $I$ or one of its Weyl reflections, and throughout the proof we assume this is the case.

Suppose $0\in\Delta$. We argue that by a suitable choice of $\gamma$, one can arrange that for all but one of the contributions, (i) $\langle\tau_{C},\gamma_{C}\rangle\geq 0$ with equality if and only if $0\in A_{C}$, (ii) $\langle\tau_{C},\gamma_{C}\rangle>\langle\tau_{C},\gamma_{I}\rangle$, where $\gamma_{I}$ is the orthogonal projection of $\gamma$ onto $I$, and (iii) $\langle\tau_{C},\sigma_{C}\rangle>0$. The one special contribution is denoted $m_{C_{I}}$ below and corresponds to the subspace $A_{C_{I}}=I$. By property (b), (i) and (iii) imply that for $C\neq C_{I}$, the support of $m_{C}(k,-)$ lies outside $kH_{C}$ where $H_{C}$ is the half-space

[TABLE]

Let $\mathfrak{p}$ be the intersection of $I$ with all of the half-spaces $H_{C}$ for $C\neq C_{I}$. By (ii), the relative interior of $\mathfrak{p}$, viewed as a polytope in $I$, contains the point $\gamma_{I}$, hence in particular is non-empty. By construction $m\upharpoonright C_{\mathfrak{p}}=m_{C_{I}}\upharpoonright C_{\mathfrak{p}}$. Then property (a) implies that $m_{C_{I}}$ is quasi-polynomial on $C_{\mathfrak{p}}$, hence the result.

We claim that one can ensure (i) holds for all $C$ by choosing $\gamma\in\mathfrak{t}^{\ast}_{+}$ sufficiently close to [math]. Indeed let $A_{C}^{0}$ be the subspace parallel to $A_{C}$, and let $a_{C}\in A_{C}$ be the nearest point in $A_{C}$ to [math]. Then $\gamma_{C}-a_{C}\in A_{C}^{0}$ while $\tau_{C}$, $a_{C}$ are both orthogonal to $A_{C}^{0}$, hence $\langle\tau_{C},\gamma_{C}-a_{C}\rangle=0=\langle a_{C},\gamma_{C}-a_{C}\rangle$. These imply $\langle\tau_{C},\gamma_{C}\rangle=\|a_{C}\|^{2}-\langle a_{C},\gamma\rangle$. If $0\in A_{C}$ then $a_{C}=0$ and this vanishes. Otherwise we can ensure $\langle\tau_{C},\gamma_{C}\rangle>0$ by choosing $\|\gamma\|<\|a_{C}\|$. Since only finitely many $C$ occur, we can choose $\gamma$ such that this holds for all $C$ with $0\notin A_{C}$. We now turn to verifying (ii), (iii), and also handle the case $0\notin\Delta$ along the way.

Suppose $\gamma_{C}\in\mu_{\mathfrak{g}}(C)$, so that $m_{C}$ indeed appears in (2.3). If $\alpha\in\mathcal{R}_{+}$ and $\langle\tau_{C},\alpha\rangle>0$, then since $\gamma\in\mathfrak{t}^{\ast}_{+}$ it follows that $\langle\gamma_{C},\alpha\rangle>0$. It is a consequence of the cross-section theorem (cf. [5]) that $\alpha|_{\mathfrak{t}_{C}}$ appears in the list of weights $\textnormal{wt}(\nu_{C})$. Hence

[TABLE]

where $\mathcal{R}_{+}^{\tau_{C}}$ denotes the set of positive roots $\alpha$ such that $\langle\tau_{C},\alpha\rangle>0$, and $\textnormal{wt}(\nu_{C})-\mathcal{R}_{+}^{\tau_{C}}$ denotes the list of weights on $\nu_{C}$ with one copy of $\alpha|_{\mathfrak{t}_{C}}$ removed for each $\alpha\in\mathcal{R}_{+}$ satisfying $\langle\tau_{C},\alpha\rangle>0$. Hence

[TABLE]

and the inequality is strict if at least one weight $\delta$ contributes in (2.5).

If $0\notin\Delta$ then, choosing $\gamma$ sufficiently close to [math], we can ensure that for each $C$ such that $0\in A_{C}$ we have $\gamma_{C}\notin\mu_{\mathfrak{g}}(M)$ (a fortiori $\gamma_{C}\notin\mu_{\mathfrak{g}}(C)$), hence $m_{C}$ does not appear in (2.3) at all. On the other hand, by (i), (2.6) and property (b), if $0\notin A_{C}$ then $m_{C}(k,0)=0$ for all $k\geq 1$. We conclude that if $0\notin\Delta$ then $m(k,0)=0$ for all $k\geq 1$.

We turn to the case $0\in\Delta\subset I$. In this case we may choose $\gamma$ such that it is simultaneously close to [math] and arbitrarily close to $\gamma_{I}$, the orthogonal projection of $\gamma$ onto $I$. Since $\tau_{C}=\gamma_{C}-\gamma$, $\langle\tau_{C},\gamma\rangle\leq\langle\tau_{C},\gamma_{C}\rangle$ with equality if and only if $\gamma_{C}=\gamma$. By taking $\gamma$ sufficiently close to $I$, one can ensure that $\langle\tau_{C},\gamma_{I}\rangle\leq\langle\tau_{C},\gamma_{C}\rangle$ with equality if and only if $\gamma_{C}=\gamma_{I}$.

We first consider contributions from components $C\in\textnormal{Comp}_{T}(M)$ such that $\gamma_{C}\notin\mathfrak{t}_{+}^{\ast}$. In this case there exists a negative root $\alpha\in\mathcal{R}_{-}$ such that $\langle\gamma_{C},\alpha\rangle>0$. It follows from the cross-section theorem that $\alpha|_{\mathfrak{t}_{C}}\in\textnormal{wt}(\nu_{C})$. Since $\gamma\in\mathfrak{t}_{+}^{\ast}$, $\langle\gamma,\alpha\rangle\leq 0$ and so

[TABLE]

As $\alpha\notin\mathcal{R}_{+}$, we see that $\delta=\alpha$ indeed contributes in (2.5), hence $\langle\tau_{C},\sigma_{C}\rangle>0$. Moreover since $\gamma_{C}\notin\mathfrak{t}_{+}^{\ast}$, $\gamma_{C}\neq\gamma_{I}$, hence $\langle\tau_{C},\gamma_{I}\rangle<\langle\tau_{C},\gamma_{C}\rangle$. This establishes (ii), (iii) for this case.

We are left to consider contributions from $C\in\textnormal{Comp}_{T}(M)$ such that $\gamma_{C}\in\Delta=\mu_{\mathfrak{g}}(M)\cap\mathfrak{t}_{+}^{\ast}$. Let $\Delta_{\textnormal{reg}}\subset\Delta$ be the relatively open dense subset of weakly regular values. The connected components of $\Delta_{\textnormal{reg}}$ are relatively open polytopes inside the subspace $I$. Choose a connected component $\mathfrak{a}\subset\Delta_{\textnormal{reg}}$ containing [math] in its closure. We may choose $\gamma\in\mathfrak{t}^{\ast}_{+}$ such that the orthogonal projection $\gamma_{I}$ onto $I$ lies in $\mathfrak{a}$. The fibre $\mu_{\mathfrak{g}}^{-1}(\gamma_{I})$ is connected and contained in $M^{T_{I}}$, hence there is a unique connected component $C_{I}\subset M^{T_{I}}$ containing $\mu_{\mathfrak{g}}^{-1}(\gamma_{I})$. Then $A_{C_{I}}=I$ and by property (a), $m_{C_{I}}$ is quasi-polynomial on the set of integral points in $\bm{A}_{C}=\{(t,t\tau)\,|\,t>0,\,\tau\in I\}\supset C_{\mathfrak{p}}$.

The final situation to consider consists of the contributions from $C\in\textnormal{Comp}_{T}(M)$ such that $\gamma_{C}\in\Delta\setminus\Delta_{\textnormal{reg}}$. In particular $\gamma_{C}\neq\gamma_{I}$ hence

[TABLE]

establishing (ii) for this case. Let $\sigma$ be the face of $\mathfrak{t}^{\ast}_{+}$ containing $\gamma_{C}$. The subset

[TABLE]

where the union is taken over relatively open faces of $\mathfrak{t}^{\ast}_{+}$ whose closure contains $\sigma$, is a slice for the coadjoint $G_{\sigma}$-action. Let $Y=\mu_{\mathfrak{g}}^{-1}(U)$ be the corresponding symplectic cross-section, cf. [5, Remark 3.7, Theorem 3.8]. Consider the function $f=\langle\tau_{C},\mu\rangle|_{Y}\colon Y\rightarrow\mathbb{R}$, for which $C\cap Y\subset Y^{\tau_{C}}=\textnormal{Crit}(f)$ is a critical submanifold. Note that $f|_{C\cap Y}=\langle\tau_{C},\gamma_{C}\rangle$. A result from symplectic geometry says that in a suitable tubular neighborhood of $C\cap Y$, the function $f$ takes the form

[TABLE]

where $\delta_{j}\in\textnormal{wt}(\nu_{C\cap Y,Y})$, $\langle\tau_{C},\delta_{j}\rangle\neq 0$, $z_{j}$ is a vector in the subbundle of $\nu_{C\cap Y,Y}$ where $\mathfrak{t}_{C}$ acts with weight $\delta_{j}$, and $|z_{j}|$ denotes its norm with respect to a suitable Hermitian structure.

Let $S$ be the line segment with endpoints $\gamma_{I}$ and $\gamma_{C}$. By convexity $S\subset\Delta$. The inverse image $\mu_{\mathfrak{g}}^{-1}(S)\subset Y$ is connected since $\mu_{\mathfrak{g}}$ has connected fibres. By (2.7), along the line segment $S$, $f$ varies between its absolute minimum $\langle\tau_{C},\gamma_{I}\rangle$ on the fibre $\mu_{\mathfrak{g}}^{-1}(\gamma_{I})$ and its absolute maximum $\langle\tau_{C},\gamma_{C}\rangle$ on the fibre $\mu_{\mathfrak{g}}^{-1}(\gamma_{C})$. By connectedness of $\mu_{\mathfrak{g}}^{-1}(S)$ and equation (2.8), there must exist a $\delta_{j}$ such that $\langle\tau_{C},\delta_{j}\rangle>0$.

By the cross-section theorem $\nu_{Y,M}|_{C\cap Y}\simeq(C\cap Y)\times\mathfrak{g}_{\gamma_{C}}^{\perp}$, where the orthogonal complement $\mathfrak{g}_{\gamma_{C}}^{\perp}$ is embedded in $TM|_{C\cap Y}$ as the orbit directions. The weights $\mathcal{R}_{+}^{\tau_{C}}$ which are removed in (2.5) can be identified with the weights of the $\mathfrak{t}_{C}$-action on $\nu_{Y,M}|_{C\cap Y}$. With this understanding we have $\textnormal{wt}(\nu_{C\cap Y,Y})\subset\textnormal{wt}(\nu_{C})-\mathcal{R}_{+}^{\tau_{C}}$. Thus $\delta_{j}$ indeed contributes to (2.5), establishing (iii) for this case. This completes the proof. ∎

Corollary 2.3**.**

Suppose $0\in\Delta$ and let $\mathfrak{p}\subset\Delta$ be as in Theorem 2.2. If $\xi\in\mathfrak{p}$ is rational and $n_{\xi}\in\mathbb{Z}_{>0}$ is the least positive integer such that $n_{\xi}\xi\in\Lambda$, then the function

[TABLE]

is quasi-polynomial. Moreover $m\upharpoonright C_{\mathfrak{p}}$ is the unique quasi-polynomial function such that $m(k,k\xi)\allowbreak=f_{\xi}(k)$ for all rational, weakly regular values $\xi$ in the relative interior of $\mathfrak{p}$.

Remark 2.4**.**

A suitable finite collection of the functions $f_{\xi}$ already fully determines $m\upharpoonright C_{\mathfrak{p}}$.

3 Stationary phase calculation

Assume $0\in\Delta$ and let $\mathfrak{p}\subset\Delta$ be as in Theorem 2.2, so that $m\upharpoonright C_{\mathfrak{p}}$ is quasi-polynomial. By Corollary 2.3, $m\upharpoonright C_{\mathfrak{p}}$ is completely determined by the collection of quasi-polynomial functions $f_{\xi}(k)=m(k,k\xi)$, for $\xi$ ranging over rational, weakly regular values of $\mu_{\mathfrak{g}}$ lying in the relative interior of $\mathfrak{p}$. In this section we use the Berline–Vergne index formula and the stationary phase expansion to compute the functions $f_{\xi}$, and hence also $m\upharpoonright C_{\mathfrak{p}}$. The end result will be the formula (1.3) in Theorem 1.2.

Let $t\in T$. By the Berline–Vergne formula, for $X\in\mathfrak{t}$ sufficiently small one has Q\big{(}k,t{\rm e}^{X}\big{)}=Q_{t}(k,X) where

[TABLE]

and \textnormal{Td}\big{(}M^{t},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}, \mathcal{D}_{\mathbb{C}}^{t}\big{(}\nu_{M^{t},M},\tfrac{2\pi}{{\mathrm{i}}}X\big{)} denote equivariant extensions of the usual Chern-Weil forms, closed with respect to the differential ${\rm d}+2\pi{\mathrm{i}}\iota(X_{M})$, obtained by replacing curvatures with equivariant curvatures (evaluated at $\tfrac{2\pi}{{\mathrm{i}}}X$) in the usual formulas (cf. [1] for details, although note that we are using the topologist’s convention for characteristic classes).

Let $B_{r}$ denote the ball of radius $r>0$ around the origin in $\mathfrak{g}/\mathfrak{t}$. Let $\mu_{\mathfrak{g}/\mathfrak{t}}$ denote the composition of $\mu_{\mathfrak{g}}$ with the quotient map $\mathfrak{g}\rightarrow\mathfrak{g}/\mathfrak{t}$. Let $\mathfrak{g}^{t}\subset\mathfrak{g}$ denote the fixed-point set of $\textnormal{Ad}_{t}$. Then $B_{r}^{t}$ is a neighborhood of [math] in $\mathfrak{g}^{t}/\mathfrak{t}$. Recall $\mathfrak{g}^{t}/\mathfrak{t}$, $\mathfrak{g}/\mathfrak{g}^{t}$ are equipped with complex structures such that their $+{\mathrm{i}}$-eigenspaces are identified with sums of positive root spaces. Equip $\mathfrak{g}^{t}/\mathfrak{t}$ with the orientation induced by the complex structure, and let $\tau_{\mathfrak{g}^{t}/\mathfrak{t}}(X)$ be a $T$-equivariant Thom form with support contained in $B_{r}^{t}$, closed for the differential ${\rm d}-\iota(X_{M})$. Consider the $T$-equivariant differential form on $\mathfrak{g}^{t}/\mathfrak{t}$ (closed for the differential ${\rm d}+2\pi{\mathrm{i}}\iota(X_{\mathfrak{g}^{t}/\mathfrak{t}})$) given by

[TABLE]

The map $\mu_{\mathfrak{g}/\mathfrak{t}}$ restricts to a map $M^{t}\rightarrow\mathfrak{g}^{t}/\mathfrak{t}$, which we use to pull back the form \textnormal{Ch}^{t}\big{(}{\sf b},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}.

Lemma 3.1**.**

[TABLE]

Proof.

The pullback of $\tau_{\mathfrak{g}^{t}/\mathfrak{t}}(X)$ to $0\in\mathfrak{g}^{t}/\mathfrak{t}$ is the equivariant Euler class, which (since [math] is just a point) is the function

[TABLE]

where $\mathcal{R}^{\mathfrak{g}^{t}}_{+}\subset\mathcal{R}_{+}$ is a set of positive roots for $\mathfrak{g}^{t}$. Note also that $t$ acts trivially on $\mathfrak{g}^{t}/\mathfrak{t}$, since $\mathfrak{g}^{t}$ is the fixed point subspace under the adjoint action. It follows that the pullback of \textnormal{Ch}^{t}\big{(}{\sf b},\tfrac{2\pi}{{\mathrm{i}}}X\big{)} to $0\in\mathfrak{g}^{t}/\mathfrak{t}$ is the function \textnormal{det}_{\mathbb{C}}^{\mathfrak{g}/\mathfrak{t}}\big{(}1-t^{-1}{\rm e}^{-X}\big{)}. Since pullback to $\{0\}=(\mathfrak{g}^{t}/\mathfrak{t})^{T}$ is injective on equivariant cohomology classes, \textnormal{Ch}^{t}\big{(}{\sf b},\tfrac{2\pi}{{\mathrm{i}}}X\big{)}, \textnormal{det}_{\mathbb{C}}^{\mathfrak{g}/\mathfrak{t}}\big{(}1-t^{-1}{\rm e}^{-X}\big{)} determine the same class in $T$-equivariant cohomology of $\mathfrak{g}^{t}/\mathfrak{t}$. As $M$ is compact, we may make this replacement in (3.1) without changing the value of the integral. ∎

Remark 3.2**.**

The reason for the notation is that \textnormal{Ch}^{t}\big{(}{\sf b},\tfrac{2\pi}{{\mathrm{i}}}X\big{)} is a representative for the $t$-twisted Chern character of a Bott element ${\sf b}\in K_{T}^{0}(\mathfrak{g}/\mathfrak{t})$, which generates the latter as an $R(T)=K_{T}^{0}(\textnormal{pt})$-module. To be more precise, ${\sf b}$ is the generator whose pullback to $0\in\mathfrak{g}/\mathfrak{t}$ is $[\wedge^{\textnormal{ev}}\mathfrak{n}_{-}]-[\wedge^{\textnormal{odd}}\mathfrak{n}_{-}]\in K^{0}_{T}(\textnormal{pt})$, $\mathfrak{n}_{-}$ being the direct sum of the negative root spaces.

Since $T$ is compact, there exists a finite set $S\subset T$ and an open cover $\{U_{t}\,|\,t\in S\}$ of $T$ where $U_{t}$ is a small open ball around $t$ in $T$ such that Q\big{(}k,t{\rm e}^{X}\big{)}=Q_{t}(k,X) for $t{\rm e}^{X}\in U_{t}$. Let $\sigma_{t}$, $t\in S$ be bump functions on $\mathfrak{t}$ such that $\{\hat{t}_{\ast}\sigma_{t}\,|\,t\in S\}$ is a partition of unity subordinate to the cover, where $\hat{t}$ is the map

[TABLE]

which we may assume restricts to a diffeomorphism of a small ball around $0\in\mathfrak{t}$ onto $U_{t}$. By equations (3.1) and (3.2)

[TABLE]

The multiplicity function $m$ is the Fourier transform of $Q$:

[TABLE]

To do the stationary phase calculation (for $k\rightarrow\infty$) following the approach outlined at the beginning of this section, we now set $\lambda=k\xi$ where $\xi\in(\Lambda\otimes\mathbb{Q})\cap\mathfrak{p}$ is a rational, weakly regular value of $\mu_{\mathfrak{g}}$ contained in the relative interior of $\mathfrak{p}\subset\Delta$ as in Corollary 2.3, $k\in n_{\xi}\mathbb{Z}_{>0}$ and $n_{\xi}$ is the least positive integer such that $n_{\xi}\xi\in\Lambda$. Thus

[TABLE]

Let $f(m,X)=\langle\mu(m)-\xi,X\rangle$ viewed as a real-valued function on \mu_{\mathfrak{g}/\mathfrak{t}}^{-1}\big{(}B_{r}^{t}\big{)}\times\mathfrak{t}. According to the principle of stationary phase, we can include a bump function supported in a small neighborhood of the critical set of $f$ in the integrand of (3.3), and the error will be $o(k^{-\infty})$. The derivative

[TABLE]

and in particular $\textnormal{Crit}(f)\subset\mu^{-1}(\xi)\times\mathfrak{t}$. Let $\chi$ be the pullback by $\mu$ of a bump function in $\mathfrak{t}^{\ast}$ supported in a small neighborhood of $\xi$. Thus

[TABLE]

where $\sim$ denotes equality modulo an $o(k^{-\infty})$ error.

Let $Y=\mu_{\mathfrak{g}}^{-1}(\sigma)$ be the cross-section for the principal face. By the cross-section theorem, a neighborhood $N$ of $Y$ in $M$ is $G_{\sigma}$-equivariantly diffeomorphic to

[TABLE]

where $\mathfrak{g}/\mathfrak{g}_{\sigma}\simeq\mathfrak{g}_{\sigma}^{\perp}$ is embedded in the orbit directions. Since $\mu^{-1}(\xi)\cap\mu_{\mathfrak{g}}^{-1}(\mathfrak{t}^{\ast})=\mu_{\mathfrak{g}}^{-1}(\xi)\subset Y$, by taking $r$ and $\textnormal{supp}(\chi)$ sufficiently small, we can assume that $\textnormal{supp}(\chi)\cap\mu_{\mathfrak{g}/\mathfrak{t}}^{-1}(B_{r})$ is contained in a small neighborhood of $\mu_{\mathfrak{g}}^{-1}(\xi)$ where the local model $Y\times\mathfrak{g}/\mathfrak{g}_{\sigma}$ is valid, and so we may replace \mu_{\mathfrak{g}/\mathfrak{t}}^{-1}\big{(}B_{r}^{t}\big{)} with $N^{t}$ in equation (3.4). In the next lemma we use the Thom form to integrate over the $(\mathfrak{g}/\mathfrak{g}_{\sigma})^{t}$ directions.

Lemma 3.3**.**

[TABLE]

Proof.

The neighborhood $N^{t}$ of $Y^{t}$ in $M^{t}$ is $T$-equivariantly diffeomorphic to

[TABLE]

where $\mathfrak{g}_{\sigma}^{t}=(\mathfrak{g}_{\sigma})^{t}$ is the subspace of $\mathfrak{g}_{\sigma}$ fixed by $t$. Moreover the almost complex structure on $N^{t}$ is homotopic to a product almost complex structure, where $Y^{t}$ is equipped with an almost complex structure compatible with the symplectic form in the cross-section, and $\mathfrak{g}^{t}/\mathfrak{g}_{\sigma}^{t}$ is equipped with the almost complex structure whose $+{\mathrm{i}}$-eigenspace is identified with a sum of positive root spaces. Let

[TABLE]

denote the projection. For the normal bundle

[TABLE]

and again the almost complex structure is homotopic to a product one, using a compatible almost complex structure on the symplectic vector bundle $\nu_{Y^{t},Y}$, and an almost complex structure on \mathfrak{g}/\big{(}\mathfrak{g}^{t}+\mathfrak{g}_{\sigma}\big{)} whose $+{\mathrm{i}}$-eigenspace is identified with a sum of positive root spaces. Using the identifications above we obtain, up to equivariantly exact forms:

[TABLE]

Since $Y^{t}\subset\mu_{\mathfrak{g}}^{-1}(\mathfrak{t}^{\ast})$, the pullback of the equivariant Thom form $\tau_{\mathfrak{g}^{t}/\mathfrak{t}}(X)$ to $Y^{t}$ is just the function

[TABLE]

We recognize \textnormal{det}_{\mathbb{C}}^{\mathfrak{g}^{t}/\mathfrak{g}_{\sigma}^{t}}\big{(}\tfrac{{\mathrm{i}}}{2\pi}X\big{)} as the equivariant Euler class of the trivial bundle $Y^{t}\times\mathfrak{g}^{t}/\mathfrak{g}_{\sigma}^{t}$. Thus up to an equivariantly exact form, we have

[TABLE]

where $\tau_{p}(X)$ is an equivariant Thom form for the vector bundle $p\colon N^{t}=Y^{t}\times\mathfrak{g}^{t}/\mathfrak{g}_{\sigma}^{t}\rightarrow Y^{t}$.

We next want to make the replacements (3.6), (3.7) in equation (3.4), and then use the Thom form $\tau_{p}(X)$ to integrate over the fibres of $p\colon N^{t}\rightarrow Y^{t}$. In the integral over $N^{t}$ in (3.4), the integrand has compact support and all terms in the integrand are equivariantly closed except for the bump function $\chi$. By Stokes’ theorem, replacing a form by a cohomologous form in the integrand leads to an error term containing ${\rm d}\chi$; but ${\rm d}\chi$ vanishes near $\mu^{-1}(\xi)$, so the principle of stationary phase implies the error will be $o(k^{-\infty})$. Let $\iota_{Y^{t}}\colon Y^{t}\hookrightarrow N^{t}$ denote the inclusion. Similarly the formula $p_{\ast}(\tau_{p}(X)\alpha(X))=\iota_{Y^{t}}^{\ast}\alpha(X)$ applies when $\alpha(X)$ is equivariantly closed. But writing $\chi=1-(1-\chi)$, the principle of stationary phase again shows that we can make this replacement up to an $o(k^{-\infty})$ error term.

After making these replacements and integrating over the fibre, the form \tau_{p}\big{(}\tfrac{2\pi}{{\mathrm{i}}}X\big{)} disappears. There are various Lie theoretic factors left over:

[TABLE]

which simplify to \textnormal{det}_{\mathbb{C}}^{\mathfrak{g}_{\sigma}/\mathfrak{t}}\big{(}1-t^{-1}{\rm e}^{-X}\big{)} (one uses that $t$ acts trivially on $\mathfrak{g}_{\sigma}^{t}/\mathfrak{t}$ and that \big{(}\mathfrak{g}^{t}+\mathfrak{g}_{\sigma}\big{)}/\mathfrak{g}^{t}\simeq\mathfrak{g}_{\sigma}/\mathfrak{g}_{\sigma}^{t}). ∎

Choose a complementary subtorus $T_{I}^{\prime}$ so that $T\simeq T_{I}\times T_{I}^{\prime}$. The quotient map $T\rightarrow T/T_{I}$ induces an isomorphism of groups $T_{I}^{\prime}\xrightarrow{\sim}T/T_{I}$. By adding additional points if necessary, we may assume the finite subset $S\subset T$ is a product $S_{I}\times S_{I}^{\prime}$, where $S_{I}\subset T_{I}$, $S_{I}^{\prime}\subset T_{I}^{\prime}$ and that the image of $S_{I}^{\prime}$ in $T/T_{I}$ contains the set $S_{P}$ from the introduction. Thus we will write elements of $S$ as products $hg$ with $h\in S_{I}\subset T_{I}$ and $g\in S_{I}^{\prime}\subset T_{I}^{\prime}$. We may assume the bump function $\sigma_{t}$ is a product $\sigma_{h}\cdot\sigma_{g}$, where $\sigma_{h}$ (resp. $\sigma_{g}$) is a bump function on $\mathfrak{t}_{I}$ (resp. $\mathfrak{t}_{I}^{\prime}$), satisfying

[TABLE]

The next lemma gives a further simplification of (3.5).

Lemma 3.4**.**

[TABLE]

Proof.

As $T_{I}$ acts trivially on $Y$ and $\mu(Y)\subset I$, the characteristic forms in (3.5) only depend on the component of $X$ (resp. $t$) in $\mathfrak{t}_{I}^{\prime}$ (resp. $T_{I}^{\prime}$). Likewise as $\xi\in(\Lambda\otimes\mathbb{Q})\cap I$, $t^{-k\xi}$ only depends on the component $g$ of $t$ in $T_{I}^{\prime}$. This means the following expression can be split off from (3.5) and evaluated separately:

[TABLE]

The determinant is given by a product:

[TABLE]

When the product over $\mathcal{R}_{+}^{\mathfrak{g}_{\sigma}}$ is expanded, we obtain an alternating sum of terms of the form $h^{-\zeta}g^{-\zeta}{\rm e}^{-2\pi{\mathrm{i}}\langle\zeta,X\rangle}$, where $\zeta$ is a sum of a subset of $\mathcal{R}_{+}^{\mathfrak{g}_{\sigma}}$. The elements of $\mathcal{R}_{+}^{\mathfrak{g}_{\sigma}}$ lie in $\textnormal{ann}(\mathfrak{z}_{\sigma})$, the annihilator of $\mathfrak{z}_{\sigma}$ in $\mathfrak{t}^{\ast}$. Since $\mathfrak{t}^{\ast}=\mathfrak{z}_{\sigma}^{\ast}\oplus\textnormal{ann}(\mathfrak{z}_{\sigma})$ and $I\subset\mathfrak{z}_{\sigma}^{\ast}$, it follows that either $\zeta=0$ or else $\zeta\notin I$.

We claim that if $\zeta\neq 0$, then the corresponding contribution to (3.10) is [math]. Indeed taking the Fourier transform of the first equation in (3.8), we find that for any $[\zeta]\in\Lambda/(\Lambda\cap I)$, the weight lattice of $T_{I}$, we have

[TABLE]

where $\delta_{0}$ is the function on $\Lambda/(\Lambda\cap I)$ equal to $1$ at [math] and [math] otherwise, obtained by Fourier transform of the constant function $1$ on $T_{I}$. Thus for $\zeta\in\Lambda$,

[TABLE]

where $\delta_{\Lambda\cap I}$ is the function on $\Lambda$ equal to $1$ on $\Lambda\cap I$ and [math] otherwise. In particular if $\zeta\notin I$ we see that the corresponding contribution in (3.10) vanishes.

On the other hand, using equation (3.8), the contribution from $\zeta=0$ to (3.10) is

[TABLE]

This yields the expression on the right-hand-side of (3.9). ∎

We can now complete the proof of Theorem 1.2. The fibre $P=\mu^{-1}(\xi)\subset Y$ is smooth, and the quotient $\Sigma_{e}:=M_{\xi}=P/G_{\sigma}=P/T_{I}^{\prime}$ is an orbifold ($T_{I}^{\prime}$ acts locally freely on $P$). By the coisotropic embedding theorem, a neighborhood of $P$ in $Y$ is $T$-equivariantly symplectomorphic to

[TABLE]

where $B_{I}$ is a small ball around $\xi$ in the subspace $I\subset\mathfrak{t}^{\ast}$, the moment map $\mu$ is projection to the second factor, and the symplectic form

[TABLE]

where $\omega_{\xi}$ is the pullback of the symplectic form on the reduced space $M_{\xi}$, $\eta$ is the variable in $B_{I}$, $\theta\in\Omega^{1}(P,\mathfrak{t}^{\ast})^{T}$ is a connection on $P$ with curvature $F_{\theta}={\rm d}\theta$, and here as well as below we have omitted pullbacks from the notation. A neighborhood of $P^{g}$ in $Y^{g}$ is $T$-equivariantly symplectomorphic to

[TABLE]

and $T_{I}^{\prime}$ acts locally freely on $P^{g}$, with the quotient $\Sigma_{g}=P^{g}/T_{I}^{\prime}$ being an orbifold. On the same neighborhood we have

[TABLE]

Below we will omit $\textnormal{pr}_{1}^{\ast}$ from the notation.

Take the bump function $\chi$ to have its support contained in the neighborhood of $P$ where the above local normal forms are valid. We may then integrate over $I$ instead of $B_{I}$, since $\chi$ vanishes outside of $P\times B_{I}$ by assumption. On $\textnormal{supp}(\chi)$,

[TABLE]

Only the top degree part of ${\rm e}^{k\langle{\rm d}\eta,\theta\rangle}$ contributes to the integral over $I$; this top degree part is $(-1)^{n(n-1)/2}k^{n}{\rm d}\eta\cdot\Theta$, where $n=\textnormal{dim}(I)$, ${\rm d}\eta=\Pi{\rm d}\eta^{a}$, $\Theta=\Pi\theta_{a}$ in terms of coordinates on $I$. The sign $(-1)^{n(n-1)/2}$ relates the symplectic and product orientations for $P^{g}\times I$, so will be absorbed when we use Fubini’s theorem to write the integral over $P^{g}\times I$ as an iterated integral. Let $\overline{\chi}(\eta)=\chi(\eta+\xi)$, a bump function on $I$ supported near [math]. Making these substitutions, as well as a change of variables $\eta\leadsto\eta+\xi$ in the integral over $I$, the asymptotic expression (3.9) for $m(k,k\xi)$ simplifies to

[TABLE]

We need the following special case of the stationary phase expansion.

Proposition 3.5** (stationary phase expansion, cf. [4, Lemma 7.7.3]).**

Let $u(X,\eta)$ be a Schwartz function. We have the following asymptotic expansion in $k$:

[TABLE]

Remark 3.6**.**

To obtain the expression here from the expression appearing in loc. cit., one sets $x=(X,\eta)\in\mathbb{R}^{2n}$ and $A(X,\eta)=(\eta,X)$. Note also that in Hörmander’s notation $D=-{\mathrm{i}}({\rm d}/{\rm d}x)$.

We apply this to the smooth compactly supported function

[TABLE]

Although this function depends on $k$, the dependence is quasi-polynomial, and so the expansion still applies. Since $\sigma_{g}(X)$, $\overline{\chi}(\eta)$ equal $1$ in a neighborhood of [math], they have no effect on the expansion. The $\eta$ derivatives $k^{-1}\partial_{\eta_{a}}$ operate only on the factor ${\rm e}^{k\langle\eta,F_{\theta}\rangle}$. The combined effect of the operator $\Sigma_{a}\,({\mathrm{i}}/2\pi k)\partial_{\eta_{a}}\partial_{X^{a}}$ is to replace $X$ with $({\mathrm{i}}/2\pi)F_{\theta}$, yielding the asymptotic expansion

[TABLE]

(By substituting $F_{\theta}$ for $X$ in \textnormal{Td}\big{(}P^{g},X\big{)}, $\mathcal{D}_{\mathbb{C}}^{g}(\nu_{P^{g},P},X)^{-1}$, we mean to take the Taylor expansion around $X=0$ and substitute the differential form $F_{\theta}$.) At this stage we see that the contribution of $g\in S_{I}^{\prime}$ vanishes unless $P^{g}\neq\varnothing$, so that $S_{I}^{\prime}=S_{P}$ ($S_{P}$ is as in Theorem 1.2). As the characteristic forms \textnormal{Td}\big{(}P^{g},F_{\theta}\big{)}, $\mathcal{D}_{\mathbb{C}}^{g}(\nu_{P^{g},P},F_{\theta})$ appear multiplied by the form $\Theta$, which has top degree in the $T_{I}^{\prime}$ orbit directions, we can replace these characteristic forms with their horizontal parts. Substituting $F_{\theta}$ for $X$ and taking the horizontal part is the definition of the Cartan map $\textnormal{Car}_{\theta}$ for the locally free action of $T_{I}^{\prime}$ on the space $P^{g}$, hence the result is the pullback along the map $P^{g}\rightarrow\Sigma_{g}=P^{g}/T_{I}^{\prime}$ of the form

[TABLE]

(See our remarks in the introduction regarding characteristic forms for orbifolds.) Similarly the $1^{\rm st}$ Chern form $c_{1}(L_{\Sigma})$ is obtained by applying the Cartan map to the equivariant symplectic form $\omega_{\mathfrak{t}}(X)=\omega-\langle\mu,X\rangle$, and results in $c_{1}(L_{\Sigma})=\omega_{\xi}-\langle\xi,F_{\theta}\rangle$. Hence $\textnormal{Ch}(L_{\Sigma})={\rm e}^{c_{1}(L_{\Sigma})}={\rm e}^{\omega_{\xi}-\langle\xi,F_{\theta}\rangle}$. The integral over the fibres of $P^{g}\rightarrow\Sigma_{g}$ then gives $1/d$, where $d\colon\Sigma=\sqcup\Sigma_{g}\rightarrow\mathbb{Z}$ is the locally constant function giving the size of the generic stabilizer for the $T_{I}^{\prime}\simeq T/T_{I}$ action on $\sqcup P^{g}\rightarrow\Sigma$. Equation (3.11) becomes

[TABLE]

By Corollary 2.3, $m(k,k\xi)$ is a quasi-polynomial function of $k$, hence the asymptotic expansion must be exact, or in other words, ‘$\sim$’ in equation (3.12) can be replaced with ‘$=$’. Thus setting $\lambda=k\xi$ we have

[TABLE]

The right-hand-side of equation (3.13) is quasi-polynomial in $(k,\lambda)$. Hence by Corollary 2.3, equation (3.13) holds on all of $C_{\mathfrak{p}}$ (and not only at points $(k,\lambda)$ with $\lambda=k\xi$, $\xi$ a rational, weakly regular value in the relative interior of $\mathfrak{p}$). This completes the proof of Theorem 1.2.

Acknowledgements

I thank M. Vergne and E. Meinrenken for helpful conversations. I thank the referees for helpful comments and suggestions that improved the article.