Kostant-Toda lattices and the universal centralizer

TL;DR

This paper explores the relationship between Kostant-Toda lattices and the integrable system on the universal centralizer of a complex semisimple Lie algebra, revealing a canonical embedding and deeper structural similarities.

Contribution

It establishes a canonical open embedding of a dense subset of the Kostant-Toda lattice into the universal centralizer's integrable system, highlighting their structural connections.

Findings

A canonical open embedding of a dense subset of the Kostant-Toda lattice into the universal centralizer.

Qualitative features of the integrable system on the universal centralizer are analyzed.

The study contextualizes and deepens understanding of the similarities between the two integrable systems.

Abstract

To each complex semisimple Lie algebra $\mathfrak{g}$ decorated with appropriate data, one may associate two completely integrable systems. One is the well-studied Kostant-Toda lattice, while the second is an integrable system defined on the universal centralizer $\mathcal{Z}_{\mathfrak{g}}$ of $\mathfrak{g}$. These systems are similar in that each exploits and closely reflects the invariant theory of $\mathfrak{g}$, as developed by Chevalley, Kostant, and others. One also has Kostant's description of level sets in the Kostant-Toda lattice, which turns out to suggest deeper similarities between the two integrable systems in question. We study relationships between the two aforementioned integrable systems, partly to understand and contextualize the similarities mentioned above. Our main result is a canonical open embedding of a flow-invariant open dense subset of the Kostant-Toda lattice into $\mathcal{Z}_{\mathfrak{g}}$. Secondary results include some qualitative features of the integrable system on $\mathcal{Z}_{\mathfrak{g}}$.

Equations

$${\mathcal{Z}}_{\mathfrak{g}}:=\{(g,x)\in G\times \mathcal{S}:g\in G_x\},$$

$${\tilde{f}}_i(g,x)=f_i(x),(g,x)\in {\mathcal{Z}}_{\mathfrak{g}}$$

$$\mathcal{O}_{\text{KT}}:=\xi+\mathfrak{t}+\sum_{\alpha\in\Delta}(\mathfrak{g}_{\alpha}\setminus\{0\}):=\bigg{\{}\xi+x+\sum_{\alpha\in\Delta}y_{\alpha}:x\in\mathfrak{t}\text{ and }y_{\alpha}\in\mathfrak{g}_{\alpha}\setminus\{0\}\text{ for all }\alpha\in\Delta\bigg{\}}$$

$$\displaystyle\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 6.75pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\\&&\crcr}}}\ignorespaces{\hbox{\kern-6.75pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathcal{V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 14.03464pt\raise-40.29164pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-3.0pt\hbox{$\scriptstyle{\overline{F}\big{|}_{\mathcal{V}}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 54.75pt\raise-54.65446pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 57.76532pt\raise 5.1875pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8264pt\hbox{$\scriptstyle{\varphi}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 118.65427pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 59.70213pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 118.65427pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathcal{Z}_{\mathfrak{g}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 94.327pt\raise-37.90276pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-3.61111pt\hbox{$\scriptstyle{\widetilde{F}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 70.65427pt\raise-54.73213pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-3.0pt\raise-62.5833pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 54.75pt\raise-62.5833pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathbb{C}^{r}}$}}}}}}}{\hbox{\kern 123.10982pt\raise-62.5833pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces.$$

$$\mathfrak{g}\to {\mathfrak{g}}^{\ast },x\mapsto ⟨x,\cdot ⟩,x\in \mathfrak{g}$$

$$T^{\ast }G\cong G\times {\mathfrak{g}}^{\ast }\cong G\times \mathfrak{g}$$

$$\Omega_{(g,x)}\bigg{(}(d_{e}L_{g}(y_{1}),z_{1}),(d_{e}L_{g}(y_{2}),z_{2})\bigg{)}=\langle y_{1},z_{2}\rangle-\langle y_{2},z_{1}\rangle+\langle x,[y_{1},y_{2}]\rangle,\quad y_{1},y_{2},z_{1},z_{2}\in\mathfrak{g},$$

$$h\cdot (g,x):=(hg,x),h\in G,(g,x)\in G\times \mathfrak{g},$$

$$h\cdot (g,x):=(g{h}^{-1},{\mathrm{Ad}}_h(x)),h\in G,(g,x)\in G\times \mathfrak{g}$$

$${\mu }_L:G\times \mathfrak{g}\to \mathfrak{g},(g,x)\mapsto {\mathrm{Ad}}_g(x),(g,x)\in G\times \mathfrak{g},$$

$${\mu }_R:G\times \mathfrak{g}\to \mathfrak{g},(g,x)\mapsto -x,(g,x)\in G\times \mathfrak{g}$$

$$(h_1,h_2)\cdot (g,x):=(h_{1}g{h}_2^{-1},{\mathrm{Ad}}_{h_2}(x)),(h_1,h_2)\in G\times G,(g,x)\in G\times \mathfrak{g}$$

$$\mu :=({\mu }_L,{\mu }_R):G\times \mathfrak{g}\to \mathfrak{g}\oplus \mathfrak{g},(g,x)\mapsto ({\mathrm{Ad}}_g(x),-x),(g,x)\in G\times \mathfrak{g}$$

$$\mathcal{S}:=\xi +{\mathfrak{g}}_{\eta }\subseteq \mathfrak{g}.$$

$${\mathfrak{g}}_{\text{reg}}:=\{x\in \mathfrak{g}:\dim ({\mathfrak{g}}_x)=r\}.$$

$$\pi :{\mathcal{Z}}_{\mathfrak{g}}\to \mathcal{S},(g,x)\mapsto x,(g,x)\in {\mathcal{Z}}_{\mathfrak{g}},$$

$$\dim ({\pi }^{-1}(x))=\dim (G_x)=\dim ({\mathfrak{g}}_x)=r$$

$$\mu (g,x)=({\mathrm{Ad}}_g(x),-x)=(x,-x)\in \mathcal{S}\times (-\mathcal{S}),$$

$$F\big{|}_{\mathcal{S}}:\mathcal{S}\xrightarrow{\cong}\mathbb{C}^{r},\quad x\mapsto(f_{1}(x),\ldots,f_{r}(x)),\quad x\in\mathcal{S}.$$

$${\tilde{f}}_i(g,x)=f_i(x),(g,x)\in {\mathcal{Z}}_{\mathfrak{g}},i=1,\dots ,r.$$

$$({\mu }_L^{\ast }(f_i))(g,x)=f_i({\mathrm{Ad}}_g(x))=f_i(x)={\tilde{f}}_i(g,x)$$

$$\tilde{F}:=({\tilde{f}}_1,\dots ,{\tilde{f}}_r):{\mathcal{Z}}_{\mathfrak{g}}\to {\mathbb{C}}^r,$$

$$(d_e{L}_g((d_x{f}_i{)}^{\vee }),0)\in T_g(G_x)\oplus \{0\}=T_{(g,x)}(G_x\times \{x\})\subseteq T_{(g,x)}{\mathcal{Z}}_{\mathfrak{g}}.$$

$$H(\widetilde{f_i}{)}_{(g,x)}=(d_e{L}_g((d_x{f}_i{)}^{\vee }),0).$$

$$d_{(g,x)}\widetilde{f_i}(v)=d_x{f}_i(z).$$

$$\displaystyle\omega_{(g,x)}\bigg{(}(d_{e}L_{g}((d_{x}f_{i})^{\vee}),0),v\bigg{)}$$

$$\displaystyle=\Omega_{(g,x)}\bigg{(}(d_{e}L_{g}((d_{x}f_{i})^{\vee}),0),(d_{e}L_{g}(y),z)\bigg{)}\quad\text{(by Proposition \ref{Proposition: Symplectic subvariety})}$$

$$=⟨(d_x{f}_i{)}^{\vee },z⟩+⟨x,[(d_x{f}_i{)}^{\vee },y]⟩\text{(by eqrefEquation: Symplectic form on cotangent bundle)}$$

$$=d_x{f}_i(z)+⟨[x,(d_x{f}_i{)}^{\vee }],y⟩$$

$$=d_x{f}_i(z)\text{(since (dxfi)∨∈gx)}$$

$$=d_{(g,x)}\widetilde{f_i}(v)\text{(by eqrefEquation: Preliminary)}.$$

$${\Phi }_i:\mathbb{C}\times {\mathcal{Z}}_{\mathfrak{g}}\to {\mathcal{Z}}_{\mathfrak{g}},(t,(g,x))\mapsto (g\exp (t(d_x{f}_i{)}^{\vee }),x),(t,(g,x))\in \mathbb{C}\times {\mathcal{Z}}_{\mathfrak{g}}.$$

$${\Phi }_{\lambda }:=({\Phi }_1{)}_{{\lambda }_1}\circ \cdots \circ ({\Phi }_r{)}_{{\lambda }_r}:{\mathcal{Z}}_{\mathfrak{g}}\to {\mathcal{Z}}_{\mathfrak{g}}$$

$$\Phi :{\mathbb{C}}^r\times \mathcal{S}\to {\mathcal{Z}}_{\mathfrak{g}},(\lambda ,x)\mapsto {\Phi }_{\lambda }(e,x),(\lambda ,x)\in {\mathbb{C}}^r\times \mathcal{S}.$$

Full text

We study relationships between the two aforementioned integrable systems, partly to understand and contextualize the similarities mentioned above. Our main result is a canonical open embedding of a flow-invariant open dense subset of the Kostant–Toda lattice into $\mathcal{Z}_{\mathfrak{g}}$. Secondary results include some qualitative features of the integrable system on $\mathcal{Z}_{\mathfrak{g}}$.

1. 1 Introduction

2. 1.1 Motivation and context

3. 1.2 Outline of results

4. 1.3 Organization

5. 2 Notation and conventions

6. 3 The universal centralizer

7. 3.1 Symplectic structure

8. 3.2 The integrable system

9. 4 The Kostant–Toda lattice

10. 4.1 The basics

11. 4.2 Some technical results

12. 4.3 Level sets

13. 4.4 A holomorphic map $\lambda:\mathcal{V}\rightarrow G^{*}$

14. 5 The relationship between $\mathcal{O}_{\text{KT}}$ and $\mathcal{Z}_{\mathfrak{g}}$

15. 5.1 Some preliminaries

16. 5.2 Proof of Theorem 1

1. Introduction

1.1. Motivation and context

The finite, non-periodic Toda lattice is a cornerstone of classical integrable systems theory [22, 23, 19], and it makes well-documented connections to representation theory [2, 11, 16, 10, 21]. These connections are perhaps best elucidated through Kostant’s Lie-theoretic realization of the Toda lattice [15], a completely integrable system $\overline{F}:\mathcal{O}_{\text{KT}}\rightarrow\mathbb{C}^{r}$ that one sometimes calls the Kostant–Toda lattice. Kostant defines this system for each rank-$r$ complex semisimple Lie algebra $\mathfrak{g}$ decorated with certain Lie-theoretic data, and he gives a particularly elegant description of this system’s level sets. His description involves the adjoint group $G$ of $\mathfrak{g}$, and is roughly stated as follows: each level set $\overline{F}^{-1}(z)$ admits an explicit open embedding into the $G$-stabilizer of a suitable regular element in $\mathfrak{g}$.

The preceding discussion features prominetly in [1], where (among other things) Abe and the current author speculate about a possible role to be played by the universal centralizer $\mathcal{Z}_{\mathfrak{g}}$ of $\mathfrak{g}$. This smooth symplectic variety has received some attention in representation-theoretic contexts [3, 9], and it turns out to come equipped with a completely integrable system $\widetilde{F}:\mathcal{Z}_{\mathfrak{g}}\rightarrow\mathbb{C}^{r}$. Each level set of the aforementioned system is canonically isomorphic to the $G$-stabilizer of an explicit regular element in $\mathfrak{g}$. Comparing this fact with Kostant’s description of the level sets $\overline{F}^{-1}(z)$, we are prompted to ask the following question: to what extent are the integrable systems $\overline{F}:\mathcal{O}_{\text{KT}}\rightarrow\mathbb{C}^{r}$ and $\widetilde{F}:\mathcal{Z}_{\mathfrak{g}}\rightarrow\mathbb{C}^{r}$ related?

1.2. Outline of results

Our main contribution is an answer to the question posed above. We also establish some incidental facts about the integrable system $\widetilde{F}:\mathcal{Z}_{\mathfrak{g}}\rightarrow\mathbb{C}^{r}$, facts that we believe to be independently interesting and potentially useful in other contexts.

To outline our work, we let $\mathfrak{g}$ and $G$ be exactly as described earlier and we fix the following data:

• (i)

homogeneous, algebraically independent generators $f_{1},\ldots,f_{r}$ of the invariant ring $\mathrm{Sym}(\mathfrak{g}^{*})^{G}$;

• (ii)

a Cartan subalgebra $\mathfrak{t}\subseteq\mathfrak{g}$;

• (iii)

a choice of simple roots $\Delta\subseteq\mathfrak{t}^{*}$;

• (iv)

for each $\alpha\in\Delta$, choices of $e_{\alpha}\in\mathfrak{g}_{\alpha}$ and $e_{-\alpha}\in\mathfrak{g}_{-\alpha}$ that pair to $1$ under the Killing form, where $\mathfrak{g}_{\pm\alpha}\subseteq\mathfrak{g}$ is the root space associated to $\pm\alpha$;

• (v)

an $\mathfrak{sl}_{2}$-triple $(\xi,h,\eta)\in\mathfrak{g}^{\oplus 3}$, where $h\in\mathfrak{t}$ is defined by $\alpha(h)=-2$ for all $\alpha\in\Delta$, $\xi:=\sum_{\alpha\in\Delta}e_{-\alpha}$, and $\eta:=\sum_{\alpha\in\Delta}c_{\alpha}e_{\alpha}$ for suitable coefficients $c_{\alpha}\in\mathbb{C}^{\times}$.

This $\mathfrak{sl}_{2}$-triple canonically determines a Slodowy slice (a.k.a Kostant section) $\mathcal{S}\subseteq\mathfrak{g}$, and the universal centralizer is then the following smooth, $2r$-dimensional, closed subvariety of $G\times\mathcal{S}$:

[TABLE]

where $G_{x}$ denotes the $G$-stabilizer of $x\in\mathcal{S}$ under the adjoint action. The universal centralizer has a canonical symplectic structure (Proposition 2), in which context we consider the functions $\widetilde{f}_{1},\ldots,\widetilde{f}_{r}:\mathcal{Z}_{\mathfrak{g}}\rightarrow\mathbb{C}$ defined by

[TABLE]

for all $i\in\{1,\ldots,r\}$. In particular, we note that $\widetilde{F}:=(\widetilde{f}_{1},\ldots,\widetilde{f}_{r}):\mathcal{Z}_{\mathfrak{g}}\rightarrow\mathbb{C}^{r}$ is a completely integrable system (Proposition 4) with level sets satisfying $\widetilde{F}^{-1}(\widetilde{F}(g,x))\cong G_{x}$ for all $(g,x)\in\mathcal{Z}_{\mathfrak{g}}$ (Proposition 5). This system turns out to be quite concrete, an idea that we emphasize through an explicit construction of Carathéodory–Jacobi–Lie coordinates (see the end of Section 3.2).

Now consider the $2r$-dimensional locally closed subvariety

[TABLE]

of $\mathfrak{g}$, which is known to carry a distinguished symplectic form. Let $\overline{f}_{1},\ldots,\overline{f}_{r}:\mathcal{O}_{\text{KT}}\rightarrow\mathbb{C}$ denote the restrictions of $f_{1},\ldots,f_{r}$ to $\mathcal{O}_{\text{KT}}$, respectively. Kostant [15] proves $\overline{F}:=(\overline{f}_{1},\ldots,\overline{f}_{r}):\mathcal{O}_{\text{KT}}\rightarrow\mathbb{C}^{r}$ defines a completely integrable system, hereafter called the Kostant–Toda lattice.

Our main result is the following relationship between the integrable systems $\overline{F}:\mathcal{O}_{\text{KT}}\rightarrow\mathbb{C}^{r}$ and $\widetilde{F}:\mathcal{Z}_{\mathfrak{g}}\rightarrow\mathbb{C}^{r}$.

Theorem 1**.**

There exist a canonical open dense subset $\mathcal{V}\subseteq\mathcal{O}_{\emph{KT}}$ and a canonical open embedding of complex manifolds $\varphi:\mathcal{V}\rightarrow\mathcal{Z}_{\mathfrak{g}}$ satisfying the following conditions.

• (i)

The open set $\mathcal{V}$ is a union of level sets of $\overline{F}$.

• (ii)

We have a commutative diagram

[TABLE]

• (iii)

For all $i\in\{1,\ldots,r\}$, $\varphi$ identifies the Hamiltonian vector field of $\overline{f}_{i}$ on $\mathcal{V}$ with the Hamiltonian vector field of $\widetilde{f}_{i}$ on the image of $\varphi$.

1.3. Organization

Section 2 briefly describes some of the notation and conventions adopted throughout this paper. The main mathematical content then begins in Section 3, which concerns the universal centralizer and its completely integrable system. Section 4 subsequently studies aspects of the Kostant–Toda lattice, especially those aspects pertinent to Theorem 1. A proof of Theorem 1 is then given in Section 5.

Acknowledgements

The author wishes to thank Ana Bălibanu for enlightening conversations. He also gratefully acknowledges support from the Natural Sciences and Engineering Research Council of Canada [PDF–516638–2018].

2. Notation and conventions

This paper works exclusively over $\mathbb{C}$, viewing it as the base field of anything whose definition presupposes a base field (e.g. a variety, manifold, Lie algebra, etc.). We deal extensively with affine varieties, using $\mathbb{C}[X]$ to denote the coordinate ring of any affine variety $X$. Furthermore, we always understand “group action” as meaning “left group action”.

We make many statements concerning the openness, closedness, closure, and denseness of sets. Whenever any such statement is ambiguous about whether it uses the algebro-geometric / Zariski topology versus the Euclidean / complex-analytic topology, the latter topology is being used implicitly.

Throughout this paper, $\mathfrak{g}$ is a finite-dimensional, semisimple, rank-$r$ Lie algebra with adjoint group $G$ and associated exponential map $\exp:\mathfrak{g}\rightarrow G$. The second paragraph of 1.2 chooses the additional Lie-theoretic data (i)–(v), and we now regard these as fixed for the duration of the paper. We also have an adjoint representation $\operatorname{Ad}:G\rightarrow\operatorname{GL}(\mathfrak{g})$, $g\mapsto\operatorname{Ad}_{g}$, through which $G$ acts on $\mathfrak{g}$. Denote by $G_{x}\subseteq G$ (resp. $Gx\subseteq\mathfrak{g}$) the $G$-stabilizer (resp. $G$-orbit) of any $x\in\mathfrak{g}$, noting that the Lie algebra of $G_{x}$ is precisely $\mathfrak{g}_{x}:=\{y\in\mathfrak{g}:[x,y]=0\}$.

Let $\langle\cdot,\cdot\rangle:\mathfrak{g}\otimes_{\mathbb{C}}\mathfrak{g}\rightarrow\mathbb{C}$ be the Killing form, which one knows to be non-degenerate and $G$-invariant. It follows that

[TABLE]

defines an isomorphism between the adjoint representation and its dual. This isomorphism allows us to understand the moment maps of Hamiltonian $G$-actions as being $\mathfrak{g}$-valued, and it allows us to transfer the canonical Poisson structure on $\mathfrak{g}^{*}$ to $\mathfrak{g}$. Note that $\mathbb{C}[\mathfrak{g}]$ thereby becomes a Poisson algebra with Poisson centre $\mathbb{C}[\mathfrak{g}]^{G}$, the algebra of $G$-invariant regular functions on $\mathfrak{g}$.

3. The universal centralizer

We now formalize our introductory discussion of the universal centralizer and its properties. This begins in Section 3.1, which realizes the universal centralizer as a symplectic subvariety of $T^{*}G$ (Proposition 2). Section 3.2 is devoted to the integrable system on the universal centralizer, and includes results on level sets (Proposition 5), Hamiltonian vector fields (Proposition 6), and Carathéodory–Jacobi–Lie coordinates (Proposition 7, Proposition 8, and the last paragraph of 3.2).

3.1. Symplectic structure

Note that the left trivialization of $T^{*}G$ and the isomorphism (3) determine isomorphisms

[TABLE]

of vector bundles over $G$. The canonical symplectic form on $T^{*}G$ thereby corresponds to a symplectic form $\Omega$ on $G\times\mathfrak{g}$, defined as follows on each tangent space $T_{(g,x)}(G\times\mathfrak{g})=T_{g}G\oplus\mathfrak{g}$:

[TABLE]

where $L_{g}:G\rightarrow G$ is left translation by $g$ and $d_{e}L_{g}:\mathfrak{g}\rightarrow T_{g}G$ is the differential of $L_{g}$ at the identity $e\in G$ (cf. [18, Section 5, Equation (14L)]). It is then not difficult to verify that

[TABLE]

define commuting Hamiltonian actions of $G$ on $G\times\mathfrak{g}$, and that

[TABLE]

are moment maps for (5a) and (5b), respectively. It follows that

[TABLE]

defines a Hamiltonian action of $G\times G$ on $G\times\mathfrak{g}$, for which

[TABLE]

is necessarily a moment map.

Now recall the $\mathfrak{sl}_{2}$-triple $(\xi,h,\eta)$ fixed in 1.2, noting that the associated Slodowy slice is

[TABLE]

This slice is fruitfully studied in relation to $\mathfrak{g}_{\text{reg}}$, the open, dense, $G$-invariant subvariety of $\mathfrak{g}$ defined by

[TABLE]

One knows that $x\in\mathfrak{g}$ lies in $\mathfrak{g}_{\text{reg}}$ if and only if $x$ is $G$-conjugate to a vector in $\mathcal{S}$. In this case, $x$ is $G$-conjugate to a unique vector in $\mathcal{S}$ (see [13, Theorem 8]).

Now let the universal centralizer $\mathcal{Z}_{\mathfrak{g}}$ be as defined in (1), observing that $\mathcal{Z}_{\mathfrak{g}}$ is a closed subvariety of $G\times\mathfrak{g}$. Using the symplectic form (4), one can strengthen this observation as follows.

Proposition 2**.**

$\mathcal{Z}_{\mathfrak{g}}$* is a $2r$-dimensional symplectic subvariety of $G\times\mathfrak{g}$.*

Proof.

Consider the surjective map

[TABLE]

noting that $\pi^{-1}(x)\cong G_{x}$ for all $x\in\mathcal{S}$. We thus have

[TABLE]

for all $x\in\mathcal{S}$, where the last equality is based on the fact that $x\in\mathcal{S}\subseteq\mathfrak{g}_{\text{reg}}$. Note also that $\dim(\mathcal{S})=\dim(\mathfrak{g}_{\eta})=r$, as one knows that $\eta\in\mathfrak{g}_{\text{reg}}$ (see [12, Theorem 5.3]). Taken together, the previous two sentences imply that $\dim(\mathcal{Z}_{\mathfrak{g}})=2r$.

To prove that $\mathcal{Z}_{\mathfrak{g}}$ is a symplectic subvariety of $G\times\mathfrak{g}$, we first note that $((\xi,-\xi),(h,h),(\eta,-\eta))$ is an $\mathfrak{sl}_{2}$-triple in $\mathfrak{g}\oplus\mathfrak{g}$. The Slodowy slice associated to this triple is easily seen to be $\mathcal{S}\times(-\mathcal{S})\subseteq\mathfrak{g}\oplus\mathfrak{g}$, where $-\mathcal{S}:=\{-x:x\in\mathcal{S}\}\subseteq\mathfrak{g}$. An application of [8, Proposition 6] then shows $\mu^{-1}(\mathcal{S}\times(-\mathcal{S}))$ to be a (smooth) symplectic subvariety of $G\times\mathfrak{g}$, reducing us to proving $\mathcal{Z}_{\mathfrak{g}}=\mu^{-1}(\mathcal{S}\times(-\mathcal{S}))$.

Suppose that $(g,x)\in\mathcal{Z}_{\mathfrak{g}}$, so that $x\in\mathcal{S}$ and $\operatorname{Ad}_{g}(x)=x$. The formula (7) then gives

[TABLE]

establishing the inclusion $\mathcal{Z}_{\mathfrak{g}}\subseteq\mu^{-1}(S\times(-\mathcal{S}))$. Now assume that $(g,x)\in\mu^{-1}(\mathcal{S}\times(-\mathcal{S}))$, i.e. $\operatorname{Ad}_{g}(x)\in\mathcal{S}$ and $x\in\mathcal{S}$ (see (7)). By the discussion of $\mathcal{S}$ preceding this proposition, we must have $\operatorname{Ad}_{g}(x)=x$. It follows that $(g,x)\in\mathcal{Z}_{\mathfrak{g}}$, yielding the inclusion $\mu^{-1}(\mathcal{S}\times(-\mathcal{S}))\subseteq\mathcal{Z}_{\mathfrak{g}}$. This completes the proof. ∎

Remark 3**.**

The variety $G\times\mathfrak{g}$ is hyperkähler (see [17]), in which context the equation $\mathcal{Z}_{\mathfrak{g}}=\mu^{-1}(\mathcal{S}\times(-\mathcal{S}))$ realizes $\mathcal{Z}_{\mathfrak{g}}$ as a hyperkähler slice in $G\times\mathfrak{g}$. The study of hyperkähler slices originates in Bielawski’s work [4], and we refer the reader to [5] and [8, Section 4.2] for further details.

3.2. The integrable system

Let $f_{1},\ldots,f_{r}$ be the homogeneous, algebraically independent generators of $\mathbb{C}[\mathfrak{g}]^{G}$ fixed in 1.2, and define $F:\mathfrak{g}\rightarrow\mathbb{C}^{r}$ by $F=(f_{1},\ldots,f_{r})$. The restriction of $F$ to $\mathcal{S}$ is then a variety isomorphism (see [13, Theorem 7]), to be denoted

[TABLE]

Let us also define $\widetilde{f}_{1},\ldots,\widetilde{f}_{r}\in\mathbb{C}[\mathcal{Z_{\mathfrak{g}}}]$ by

[TABLE]

Proposition 4**.**

The functions $\widetilde{f}_{1},\ldots,\widetilde{f}_{r}$ form a completely integrable system on $\mathcal{Z}_{\mathfrak{g}}$.

Proof.

The number of functions coincides with $\frac{1}{2}\dim(\mathcal{Z}_{\mathfrak{g}})$ (by Proposition 2), so that we are reduced to verifying the following two things: $\widetilde{f}_{1},\ldots,\widetilde{f}_{r}$ Poisson-commute in pairs, and the differentials $d\widetilde{f}_{1},\ldots,d\widetilde{f}_{r}$ are linearly independent at all points in some open dense subset of $\mathcal{Z}_{\mathfrak{g}}$. Accordingly, we recall that $\mathbb{C}[\mathfrak{g}]^{G}$ is the Poisson centre of $\mathbb{C}[\mathfrak{g}]$. It follows that $f_{1},\ldots,f_{r}$ Poisson-commute in pairs. This implies that the pullback functions $\mu_{L}^{*}(f_{1}),\ldots,\mu_{L}^{*}(f_{r})$ Poisson-commute in pairs, as the moment map $\mu_{L}:G\times\mathfrak{g}\rightarrow\mathfrak{g}$ is a Poisson morphism. Since $Z_{\mathfrak{g}}$ is a symplectic subvariety of $G\times\mathfrak{g}$ (see Proposition 2), we conclude that the restricted functions \mu_{L}^{*}(f_{1})\big{|}_{\mathcal{Z}_{\mathfrak{g}}},\ldots,\mu_{L}^{*}(f_{r})\big{|}_{\mathcal{Z}_{\mathfrak{g}}}\in\mathbb{C}[\mathcal{Z}_{\mathfrak{g}}] must Poisson-commute in pairs. Note also that

[TABLE]

for all $(g,x)\in\mathcal{Z}_{\mathfrak{g}}$ and $i=1,\ldots,r$, i.e. \mu_{L}^{*}(f_{i})\big{|}_{\mathcal{Z}_{\mathfrak{g}}}=\widetilde{f}_{i} for all $i=1,\ldots,r$. These last two sentences imply that $\widetilde{f}_{1},\ldots,\widetilde{f}_{r}$ Poisson-commute in pairs.

To address the linear independence of $d\widetilde{f}_{1},\ldots,d\widetilde{f}_{r}$, we recall that (9) is an isomorphism. It follows that the differentials of f_{1}\big{|}_{\mathcal{S}},\ldots,f_{r}\big{|}_{\mathcal{S}}\in\mathbb{C}[\mathcal{S}] are linearly independent at every point of $\mathcal{S}$. Now observe that the map $\pi:\mathcal{Z}_{\mathfrak{g}}\rightarrow\mathcal{S}$ in (8) has a section, defined by sending $x\in\mathcal{S}$ to $(e,x)\in\mathcal{Z}_{\mathfrak{g}}$. We conclude that $\pi$ is a submersion. Together with our discussion of the differentials of f_{1}\big{|}_{\mathcal{S}},\ldots,f_{r}\big{|}_{\mathcal{S}}, this implies the following: \pi^{*}(f_{1}\big{|}_{\mathcal{S}}),\ldots,\pi^{*}(f_{r}\big{|}_{\mathcal{S}}) have linearly independent differentials at every point in $\mathcal{Z}_{\mathfrak{g}}$. It is also clear that \pi^{*}(f_{i}\big{|}_{\mathcal{S}})=\widetilde{f}_{i} for all $i=1,\ldots,r$, so that our proof complete. ∎

A natural next step is to examine the level sets of

[TABLE]

which are described as follows.

Proposition 5**.**

The map $\widetilde{F}$ is surjective and $\widetilde{F}^{-1}(\widetilde{F}(g,x))=G_{x}\times\{x\}$ for all $(g,x)\in\mathcal{Z}_{\mathfrak{g}}$.

Proof.

Given $z=(z_{1},\ldots,z_{r})\in\mathbb{C}^{r}$, one can use (9) to find $x\in\mathcal{S}$ such that $f_{i}(x)=z_{i}$ for all $i=1,\ldots,r$. It follows that $(e,x)\in\mathcal{Z}_{\mathfrak{g}}$ and $\widetilde{F}(e,x)=z$, establishing surjectivity.

Now fix $(g,x)\in\mathcal{Z}_{\mathfrak{g}}$, noting that the inclusion $G_{x}\times\{x\}\subseteq\widetilde{F}^{-1}(\widetilde{F}(g,x))$ is straightforward. To establish the opposite inclusion, suppose that $(g^{\prime},x^{\prime})\in\mathcal{Z}_{\mathfrak{g}}$ satisfies $\widetilde{F}(g^{\prime},x^{\prime})=\widetilde{F}(g,x)$. This is equivalent to the statement $f_{i}(x^{\prime})=f_{i}(x)$ for all $i=1,\ldots,r$, which by (9) forces $x=x^{\prime}$ to hold. We conclude that $g^{\prime}\in G_{x^{\prime}}=G_{x}$, giving $(g^{\prime},x^{\prime})\in G_{x}\times\{x\}$. The inclusion $\widetilde{F}^{-1}(\widetilde{F}(g,x))\subseteq G_{x}\times\{x\}$ now follows, and the proof is complete. ∎

In addition to motivating Proposition 5, Proposition 4 naturally leads us to study the Hamiltonian vector fields $H(\widetilde{f}_{1}),\ldots,H(\widetilde{f}_{r})$ of $\widetilde{f}_{1},\ldots,\widetilde{f}_{r}$, respectively. We begin this study by fixing $i\in\{1,\ldots,r\}$ and $x\in\mathfrak{g}$. Note that the differential $d_{x}f_{i}$ is naturally a vector in $\mathfrak{g}^{*}$, which the isomorphism (3) identifies with a vector $(d_{x}f_{i})^{\vee}\in\mathfrak{g}$. It is not difficult to verify that $(d_{x}f_{i})^{\vee}\in\mathfrak{g}_{x}$ (see [15, Proposition 1.3]), implying that $d_{e}L_{g}((d_{x}f_{i})^{\vee})\in T_{g}(G_{x})$ for all $g\in G_{x}$. If we assume that $x\in\mathcal{S}$ and $g\in G_{x}$, i.e. $(g,x)\in\mathcal{Z}_{\mathfrak{g}}$, then

[TABLE]

Proposition 6**.**

If $(g,x)\in\mathcal{Z}_{\mathfrak{g}}$ and $i\in\{1,\ldots,r\}$, then

[TABLE]

Proof.

Noting that $T_{(g,x)}\mathcal{Z}_{\mathfrak{g}}\subseteq T_{(g,x)}(G\times\mathfrak{g})=T_{g}G\oplus\mathfrak{g}$, any given $v\in T_{(g,x)}\mathcal{Z}_{\mathfrak{g}}$ must have the form $v=(d_{e}L_{g}(y),z)$ with $y,z\in\mathfrak{g}$. It then follows immediately from (10) that

[TABLE]

Now let $\omega$ and $\Omega$ denote the symplectic forms on $\mathcal{Z}_{\mathfrak{g}}$ and $G\times\mathfrak{g}$, respectively, noting that

[TABLE]

This completes the proof. ∎

The Hamiltonian vector fields $H(\widetilde{f}_{i})$ are necessarily tangent to the level sets $G_{x}\times\{x\}\cong G_{x}$ of $\widetilde{F}$, a consequence of $\widetilde{f}_{1},\ldots,\widetilde{f}_{r}$ forming an integrable system. With this in mind, Proposition 6 implies the following description of $H(\widetilde{f}_{i})$ on each level set $G_{x}$: $H(\widetilde{f}_{i})$ is the left-invariant vector field on $G_{x}$ associated to $(d_{x}f_{i})^{\vee}\in\mathfrak{g}_{x}$. In other words, the flow of $H(\widetilde{f}_{i})$ is given by

[TABLE]

Consider the biholomorphism $(\Phi_{i})_{t}:=\Phi_{i}(t,\cdot):\mathcal{Z}_{\mathfrak{g}}\rightarrow\mathcal{Z}_{\mathfrak{g}}$ for each fixed $t\in\mathbb{C}$ and $i\in\{1,\ldots,r\}$. By appealing to generalities on Hamiltonian flows or performing a direct calculation, one sees that $(\Phi_{i})_{t}$ is a symplectomorphism. It follows that

[TABLE]

defines a symplectomorphism for each $\lambda=(\lambda_{1},\ldots,\lambda_{r})\in\mathbb{C}^{r}$. Now form the map

[TABLE]

An application of (12) reveals that

[TABLE]

Proposition 7**.**

The map $\Phi:\mathbb{C}^{r}\times\mathcal{S}\rightarrow\mathcal{Z}_{\mathfrak{g}}$ is a a surjective local biholomorphism.

Proof.

We begin with a proof of surjectivity, for which we suppose that $(g,x)\in\mathcal{Z}_{\mathfrak{g}}$. Since $x\in\mathfrak{g}_{\text{reg}}$, we have an algebraic group isomorphism $G_{x}\cong(\mathbb{C}^{\times})^{p}\times\mathbb{C}^{q}$ for some $p,q\geq 0$ with $p+q=r$ (see [15, Proposition 2.4]). It follows that the restricted exponential map \exp\big{|}_{\mathfrak{g}_{x}}:\mathfrak{g}_{x}\rightarrow G_{x} is surjective. We may therefore find $y\in\mathfrak{g}_{x}$ with $\exp(y)=g$.

Now note that the differentials of $f_{1},\ldots,f_{r}$ are linearly independent at each point of $\mathfrak{g}_{\text{reg}}$ (see [13, Theorem 9]), so that $d_{x}f_{1},\ldots,d_{x}f_{r}$ are linearly independent. It follows that $(d_{x}f_{1})^{\vee},\ldots,(d_{x}f_{r})^{\vee}$ are linearly independent elements of $\mathfrak{g}_{x}$, and a dimension count then implies that $(d_{x}f_{1})^{\vee},\ldots,(d_{x}f_{r})^{\vee}$ form a basis of $\mathfrak{g}_{x}$. We may therefore find $\lambda_{1},\ldots,\lambda_{r}\in\mathbb{C}$ for which $y=\lambda_{1}(d_{x}f_{1})^{\vee}+\cdots+\lambda_{r}(d_{x}f_{r})^{\vee}$. Setting $\lambda:=(\lambda_{1},\ldots,\lambda_{r})$, an application of (14) gives

[TABLE]

We conclude that $\Phi$ is surjective.

To establish that $\Phi$ is a local biholomorphism, fix a point $(\lambda,x)\in\mathbb{C}^{r}\times\mathcal{S}$. We have

[TABLE]

so that the differential of $\Phi$ at $(\lambda,x)$ is a map

[TABLE]

Letting $e_{i}\in\mathbb{C}^{r}$ denote the $i^{\text{th}}$ standard basis vector and noting that the Hamiltonian flows $\Phi_{1},\ldots,\Phi_{r}$ necessarily commute in pairs (by Proposition 4), we have

[TABLE]

It follows that for $y=(y_{1},\ldots,y_{r})\in\mathbb{C}^{r}$,

[TABLE]

Now note that

[TABLE]

where g:=\exp\big{(}\sum_{i=1}^{r}\lambda_{i}(d_{x}f_{i})^{\vee}\big{)}. We may therefore regard $d_{(\lambda,x)}\Phi$ as taking values in $T_{g}G\oplus\mathfrak{g}$. With this in mind, the following is a straightforward consequence of (14): if $z\in\mathfrak{g}_{\eta}$, then

[TABLE]

for some vector $w\in T_{g}G$. This combines with (16) to give

[TABLE]

Suppose that $d_{(\lambda,x)}\Phi(y,z)=0$, i.e.

[TABLE]

Proposition 6 implies that the linear combination of $H(\widetilde{f}_{1})_{\Phi(\lambda,x)},\ldots,H(\widetilde{f}_{r})_{\Phi(\lambda,x)}$ lies in $T_{g}G\oplus\{0\}\subseteq T_{g}G\oplus\mathfrak{g}$, so that $z=0$ must hold. The equation (17) then gives $w=0$, and hence

[TABLE]

At the same time, the proof of Proposition 4 explains that the differentials of $\widetilde{f}_{1},\ldots,\widetilde{f}_{r}$ are linearly independent at every point in $\mathcal{Z}_{\mathfrak{g}}$. We conclude that $H(\widetilde{f}_{1}),\ldots,H(\widetilde{f}_{r})$ are linearly independent at every point in $\mathcal{Z}_{\mathfrak{g}}$, which in particular implies that $y_{1}=\cdots=y_{r}=0$.

We have shown that $(y,z)=(0,0)$, meaning that $d_{(\lambda,x)}\Phi$ is injective. Since the complex manifolds $\mathbb{C}^{r}\times\mathcal{S}$ and $\mathcal{Z}_{\mathfrak{g}}$ are both $2r$-dimensional, it follows that $d_{(\lambda,x)}\Phi$ is an isomorphism for all $(\lambda,x)\in\mathbb{C}^{r}\times\mathcal{S}$. This completes the proof. ∎

Now let $z_{1},\ldots,z_{r}$ denote the usual coordinates on $\mathbb{C}^{r}$, and recall that the \widehat{f}_{i}:=f_{i}\big{|}_{\mathcal{S}} form a system of coordinates on $\mathcal{S}$ (see (9)). Abusing notation slightly, we let $z_{1},\ldots,z_{r},\widehat{f}_{1},\ldots,\widehat{f}_{r}$ denote the induced system of coordinates on $\mathbb{C}^{r}\times\mathcal{S}$. One immediate observation is that

[TABLE]

for all $i\in\{1,\ldots,r\}$. We also note that

[TABLE]

defines a symplectic form on $\mathbb{C}^{r}\times\mathcal{S}$.

Proposition 8**.**

We have $\Phi^{*}(\omega)=\nu$, where $\omega$ is the symplectic form on $\mathcal{Z}_{\mathfrak{g}}$.

Proof.

Let $\partial f_{1},\ldots,\partial f_{r}$ denote the coordinate vector fields on $\mathcal{S}$ associated to $\widehat{f}_{1},\ldots,\widehat{f}_{r}$, respectively. It follows that for each $(\lambda,x)\in\mathbb{C}^{r}\times\mathcal{S}$, $\{(e_{1},0),\ldots,(e_{r},0),(0,(\partial f_{1})_{x}),\ldots,(0,(\partial f_{r})_{x})\}$ is a basis of $T_{(\lambda,x)}(\mathbb{C}^{r}\times\mathcal{S})=\mathbb{C}^{r}\oplus\mathfrak{g}_{\eta}$. Observe also that $\nu_{(\lambda,x)}$ is the unique skew-symmetric bilinear form on $\mathbb{C}^{r}\oplus\mathfrak{g}_{\eta}$ satisfying the following conditions:

• •

\nu_{(\lambda,x)}\big{(}(e_{i},0),(e_{j},0)\big{)}=0 for all $i,j\in\{1,\ldots,r\}$;

• •

\nu_{(\lambda,x)}\big{(}(e_{i},0),(0,(\partial f_{j})_{x})\big{)}=\delta_{ij} for all $i,j\in\{1,\ldots,r\}$;

• •

\nu_{(\lambda,x)}\big{(}(0,(\partial f_{i})_{x}),(0,(\partial f_{j})_{x})\big{)}=0 for all $i,j\in\{1,\ldots,r\}$.

It will therefore suffice to show that these identities hold after replacing $\nu_{(\lambda,x)}$ with $(\Phi^{*}(\omega))_{(\lambda,x)}$. In other words, it will suffice to verify the following identities:

• (i)

\omega_{\Phi(\lambda,x)}\big{(}d_{(\lambda,x)}\Phi(e_{i},0),d_{(\lambda,x)}\Phi(e_{j},0)\big{)}=0 for all $i,j\in\{1,\ldots,r\}$;

• (ii)

\omega_{\Phi(\lambda,x)}\big{(}d_{(\lambda,x)}\Phi(e_{i},0),d_{(\lambda,x)}\Phi(0,(\partial f_{j})_{x})\big{)}=\delta_{ij} for all $i,j\in\{1,\ldots,r\}$;

• (iii)

\omega_{\Phi(\lambda,x)}\big{(}d_{(\lambda,x)}\Phi(0,(\partial f_{i})_{x}),d_{(\lambda,x)}\Phi(0,(\partial f_{j})_{x})\big{)}=0 for all $i,j\in\{1,\ldots,r\}$.

To address (i), note that (15) gives

[TABLE]

The right hand side is necessarily zero for all $i,j\in\{1,\ldots,r\}$, as $\widetilde{f}_{1},\ldots,\widetilde{f}_{r}$ Poisson-commute in pairs (see Proposition 4). This verifies (i).

We begin the proof of (ii) with the the following calculation:

[TABLE]

where $d_{(\lambda,x)}\widehat{f}_{i}$ is the differential of $\widehat{f}_{i}$ as a coordinate on $\mathbb{C}^{r}\times\mathcal{S}$. It is straightforward to see that the last line equals $d_{x}\widehat{f}_{i}((\partial f_{j})_{x})$, where $d_{x}\widehat{f}_{i}$ is the differential of the coordinate $\widehat{f}_{i}$ on $\mathcal{S}$. Since $d_{x}\widehat{f}_{i}((\partial f_{j})_{x})=\delta_{ij}$, we have shown that (ii) holds.

Before proving (iii), we note the following straightforward consequence of the definition (13):

[TABLE]

for all $i\in\{1,\ldots,r\}$. Note that the vector $(0,(\partial f_{i})_{x})$ appearing on the right hand side belongs to $T_{(e,x)}\mathcal{Z}_{\mathfrak{g}}\subseteq T_{(e,x)}(G\times\mathfrak{g})=\mathfrak{g}\oplus\mathfrak{g}$, while on the left hand side we have $(0,(\partial f_{i})_{x})\in T_{(\lambda,x)}(\mathbb{C}^{r}\times\mathcal{S})=\mathbb{C}^{r}\oplus\mathfrak{g}_{\eta}$.

Using (19) and recalling that $\Phi_{\lambda}:\mathcal{Z}_{\mathfrak{g}}\rightarrow\mathcal{Z}_{\mathfrak{g}}$ is a symplectomorphism, we obtain

[TABLE]

Now recall that $(\mathcal{Z}_{\mathfrak{g}},\omega)$ is a symplectic subvariety of $(G\times\mathfrak{g},\Omega)$, meaning that the last line is

[TABLE]

This is zero by virtue of the formula (4), proving that (iii) holds. ∎

Remark 9**.**

Proposition 8 affords us a brief proof that $\Phi$ is a local biholomorphism. One begins by noting that $\nu$ is non-degenerate, and that the dimensions of $\mathbb{C}^{r}\times\mathcal{S}$ and $\mathcal{Z}_{\mathfrak{g}}$ coincide. Together with the identity $\Phi^{*}(\omega)=\nu$ from Proposition 8, these observations force $d_{(\lambda,x)}\Phi$ to be an isomorphism for all $(\lambda,x)\in\mathbb{C}^{r}\times\mathcal{S}$. The proof of Proposition 7 establishes this fact via more conventional arguments.

Now fix any point $x\in\mathcal{Z}_{\mathfrak{g}}$. Proposition 7 then implies that $\Phi$ restricts to a biholomorphism $U\rightarrow V$, where $V$ is a suitably chosen open neighbourhood of $x$ in $\mathcal{Z}_{\mathfrak{g}}$ and $U$ is an appropriate open subset of $\mathbb{C}^{r}\times\mathcal{S}$. The coordinates $z_{1},\ldots,z_{r},\widehat{f}_{1},\ldots,\widehat{f}_{r}$ on $\mathbb{C}^{r}\times\mathcal{S}$ thereby induce coordinates on $V$. By (18) and Proposition 8, these latter coordinates satisfy the conclusion of the Carathéodory–Jacobi–Lie theorem for completely integrable systems (e.g. [20, Theorem 4.1.8]). A less formal statement is that we have produced explicit Carathéodory–Jacobi–Lie coordinates for the integrable system on $\mathcal{Z}_{\mathfrak{g}}$.

4. The Kostant–Toda lattice

We now discuss the second integrable system featuring in our paper — the Kostant–Toda lattice. The discussion begins in 4.1, which recalls the details underlying Kostant’s construction of the Kostant–Toda lattice. Section 4.2 subsequently introduces the open set $\mathcal{V}$ from Theorem 1 (see (20)), and in addition studies certain holomorphic maps defined on $\mathcal{V}$. Using the setup developed in 4.2, Section 4.3 recalls Kostant’s description of level sets in the Kostant–Toda lattice (Theorems 16 and 17). Section 4.4 then harnesses this description to define and study a holomorphic map $\mathcal{V}\rightarrow G^{*}$ (see (25)), where $G^{*}$ is an explicit open dense subset of $G$ (see (23)).

4.1. The basics

Recall the notation and conventions set in Section 2. Note that our Cartan subalgebra $\mathfrak{t}$ and (positive) simple roots $\Delta$ determine the following data: roots $\mathcal{R}\subseteq\mathfrak{t}^{*}$, positive roots $\mathcal{R}_{+}\subseteq\mathcal{R}$, negative roots $\mathcal{R}_{-}\subseteq\mathcal{R}$, a positive Borel subalgebra $\mathfrak{b}\subseteq\mathfrak{g}$, and a negative Borel subalgebra $\mathfrak{b}_{-}\subseteq\mathfrak{g}$. Let $\mathfrak{u}$ and $\mathfrak{u}_{-}$ be the nilpotent radicals of $\mathfrak{b}$ and $\mathfrak{b}_{-}$, respectively. It follows that

[TABLE]

where

[TABLE]

is the root space associated to $\alpha\in\mathcal{R}$.

Let $T$, $B$, $B_{-}$, $U$, and $U_{-}$ denote the closed, connected subgroups of $G$ with respective Lie algebras $\mathfrak{t}$, $\mathfrak{b}$, $\mathfrak{b}_{-}$, $\mathfrak{u}$, and $\mathfrak{u}_{-}$. We then have a Weyl group $W:=N_{G}(T)/T$ and a weight lattice $\Lambda:=\mathrm{Hom}(T,\mathbb{C}^{\times})$ of all algebraic group morphisms $T\rightarrow\mathbb{C}^{\times}$. There is a canonical identification of $\Lambda$ with a $\mathbb{Z}$-lattice in $\mathfrak{t}^{*}$, and we shall make no distinction between elements of $\Lambda$ and those of the aforementioned lattice. Each $\alpha\in\mathcal{R}$ may thereby be considered an algebraic group morphism $T\rightarrow\mathbb{C}^{\times}$, allowing us to write

[TABLE]

Now let $\mathcal{O}_{\text{KT}}\subseteq\mathfrak{g}$ be as defined in (2), and let $\overline{f}_{1},\ldots,\overline{f}_{r}:\mathcal{O}_{\text{KT}}\rightarrow\mathbb{C}$ denote the restrictions of $f_{1},\ldots,f_{r}$ to $\mathcal{O}_{\text{KT}}$, respectively. With these things in mind, we summarize the results [15, Proposition 2.3.1, Proposition 6.4] and [16, Proposition 4, Theorem 29] as follows.

Theorem 10**.**

The variety $\mathcal{O}_{\emph{KT}}$ carries a canonical symplectic form, and $\overline{f}_{1},\ldots,\overline{f}_{r}$ form a completely integrable system on $\mathcal{O}_{\emph{KT}}$ with respect to this symplectic form.

We use the term Kostant–Toda lattice for the integrable system in Theorem 10. In the interest of making a simple observation about this system, we set $\overline{F}:=(\overline{f}_{1},\ldots,\overline{f}_{r}):\mathcal{O}_{\text{KT}}\rightarrow\mathbb{C}^{r}$. The observation is then as follows.

Lemma 11**.**

If $x\in\mathcal{O}_{\emph{KT}}$, then $\overline{F}^{-1}(\overline{F}(x))$ consists precisely of those points in $\mathcal{O}_{\emph{KT}}$ that are $G$-conjugate to $x$.

Proof.

Fix $x\in\mathcal{O}_{\text{KT}}$. Since $\xi+\mathfrak{b}\subseteq\mathfrak{g}_{\text{reg}}$ (see [13, Lemma 10]) and $\mathcal{O}_{\text{KT}}\subseteq\xi+\mathfrak{b}$, we have the inclusion $\mathcal{O}_{\text{KT}}\subseteq\mathfrak{g}_{\text{reg}}$. It then follows from [13, Theorem 2] that $y\in\mathcal{O}_{\text{KT}}$ is $G$-conjugate to $x$ if and only if $f_{i}(x)=f_{i}(y)$ for all $i\in\{1,\ldots,r\}$. The latter condition is precisely the statement that $\overline{F}(x)=\overline{F}(y)$, i.e. $y\in\overline{F}^{-1}(\overline{F}(x))$. This completes the proof. ∎

4.2. Some technical results

We now introduce two important subsets of $\mathfrak{t}$. The first is the open subset $\mathfrak{t}_{\text{reg}}:=\mathfrak{g}\cap\mathfrak{g}_{\text{reg}}$, which is also described by

[TABLE]

The second is the open subset of $\mathfrak{t}$ defined by

[TABLE]

Given $x\in\mathcal{C}$, one readily verifies that $\mathrm{Re}(\alpha(x))>0$ for all $\alpha\in\mathcal{R}_{+}$ and $\mathrm{Re}(\alpha(x))<0$ for all $\alpha\in\mathcal{R}_{-}$. This establishes that $\alpha(x)\neq 0$ for all $\alpha\in\mathcal{R}$, so that $\mathcal{C}\subseteq\mathfrak{t}_{\text{reg}}$.

Equipped with the preceding discussion, we may describe the image $\mathcal{D}:=F(\mathcal{C})$ of $\mathcal{C}$ under $F=(f_{1},\ldots,f_{r}):\mathfrak{g}\rightarrow\mathbb{C}^{r}$.

Lemma 12**.**

The subset $\mathcal{D}$ is open and dense in $\mathbb{C}^{r}$.

Proof.

We begin by recalling the following fundamental domain $\mathcal{C}_{\mathfrak{t}}\subseteq\mathfrak{t}$ for the action of $W$ on $\mathfrak{t}$ (see [7, Section 2.2]):

[TABLE]

Now let $\overline{\mathcal{C}}$ denote the closure of $\mathcal{C}$ in $\mathfrak{g}$, and observe that $\mathcal{C}_{\mathfrak{t}}\subseteq\overline{\mathcal{C}}$. The continuity of $F$ then implies that

[TABLE]

with the last two closures taken in $\mathbb{C}^{r}$. We also know that

[TABLE]

where the second equality is established in the proof of [13, Proposition 10] and the first equality comes from the following two things: the $W$-invariance of F\big{|}_{\mathfrak{t}} and the fact that $\mathcal{C}_{\mathfrak{t}}$ is a fundamental domain. We conclude that $\overline{\mathcal{D}}=\mathbb{C}^{r}$, i.e. $\mathcal{D}$ is dense in $\mathbb{C}^{r}$.

We now prove that $\mathcal{D}$ is open. Accordingly, note that F\big{|}_{\mathfrak{g}_{\text{reg}}}:\mathfrak{g}_{\text{reg}}\rightarrow\mathbb{C}^{r} is a submersion (see [13, Theorem 9]) whose level sets are precisely the $G$-orbits in $\mathfrak{g}_{\text{reg}}$ (see [13, Theorem 3]). It follows that $\mathrm{ker}(d_{x}F)=T_{x}(Gx)$ for all $x\in\mathfrak{g}_{\text{reg}}$. At the same time, it is not difficult to verify that $T_{x}(\mathfrak{t}_{\text{reg}})$ and $T_{x}(Gx)$ are complementary subspaces of $\mathfrak{g}$ for all $x\in\mathfrak{t}_{\text{reg}}$. These last two sentences imply that F\big{|}_{\mathfrak{t}_{\text{reg}}}:\mathfrak{t}_{\text{reg}}\rightarrow\mathbb{C}^{r} is a local biholomorphism. Since $\mathcal{C}$ is an open subset of $\mathfrak{t}_{\text{reg}}$, it follows that $F(\mathcal{C})=\mathcal{D}$ is open in $\mathbb{C}^{r}$. ∎

We now consider the open subset of $\mathcal{O}_{\mathrm{KT}}$ given by

[TABLE]

about which the following is true.

Proposition 13**.**

The open set $\mathcal{V}$ is dense in $\mathcal{O}_{\emph{KT}}$.

Proof.

It follows from [15, Proposition 2.3.1] that $\overline{F}:\mathcal{O}_{\text{KT}}\rightarrow\mathbb{C}^{r}$ is a submersion, and hence also an open map. In particular, the preimage of a dense subset under $\overline{F}$ is necessarily dense. Lemma 12 and (20) now combine to imply the desired result. ∎

We devote the balance of 4.2 to the study of two holomorphic maps on $\mathcal{V}$. The first such map involves the translated subset $\xi+\mathcal{C}\subseteq\mathfrak{g}$ and is described as follows.

Lemma 14**.**

There is a holomorphic map $\theta:\mathcal{V}\rightarrow\xi+\mathcal{C}$ with the following property: if $x\in\mathcal{V}$, then $\theta(x)$ is the unique element of $\xi+\mathcal{C}$ that is $G$-conjugate to $x$.

Proof.

By [6, Corollary 3.1.43], we have

[TABLE]

It follows that $F$ restricts to a surjective holomorphic map $F_{\mathcal{C}}:\xi+\mathcal{C}\rightarrow\mathcal{D}$, and we claim that $F_{\mathcal{C}}$ is a biholomorphism. Note that it suffices to establish injectivity.

Suppose that $x_{1},x_{2}\in\mathcal{C}$ satisfy $F(\xi+x_{1})=F(\xi+x_{2})$. It then follows from [6, Corollary 3.1.43] that $F(x_{1})=F(x_{2})$. Since $x_{1},x_{2}\in\mathfrak{g}_{\text{reg}}$, this implies that $x_{1}$ and $x_{2}$ are $G$-conjugate (see [13, Theorem 3]). Note also that $x_{1},x_{2}\in\mathfrak{t}$, so that they must actually be $W$-conjugate. At the same time, $x_{1}$ and $x_{2}$ belong to the fundamental domain $\mathcal{C}_{\mathfrak{t}}$ introduced in the proof of Lemma 12. We conclude that $x_{1}=x_{2}$, proving injectivity. As discussed above, this shows $F_{\mathcal{C}}$ to be a biholomorphism.

Let us now consider the holomorphic map

[TABLE]

Note that for all $x\in\mathcal{V}$, we have $F(\theta(x))=F(x)$. Since $x,\theta(x)\in\mathfrak{g}_{\text{reg}}$ (by [13, Lemma 10]), we conclude that $x$ and $\theta(x)$ are $G$-conjugate (see [12, Theorem 3]). Now let $y\in\xi+\mathcal{C}$ be $G$-conjugate to $x$, observing that $F(y)=F(x)=F(\theta(x))$. It follows that $y=\theta(x)$, as $F_{\mathcal{C}}$ is a bijection. This completes the proof. ∎

To construct a second holomorphic map on $\mathcal{V}$, we recall Kostant’s variety isomorphism

[TABLE]

(see [14, Theorem 1.2]). Note that $\mathcal{V}$ and $\xi+\mathcal{C}$ are both subsets of $\xi+\mathfrak{b}$, so that we may apply $\psi^{-1}$ to elements of $\mathcal{V}$ and $\xi+\mathcal{C}$. Letting $\pi_{U}:U\times\mathcal{S}\rightarrow U$ be the projection to the first factor, we define our second holomorphic map on $\mathcal{V}$ by

[TABLE]

Lemma 15**.**

If $x\in\mathcal{V}$, then $\nu(x)$ is the unique element of $U$ satisfying $\operatorname{Ad}_{\nu(x)}(\theta(x))=x$.

Proof.

Let us write $\psi^{-1}(x)=(u_{1},\overline{x})$ and $\psi^{-1}(\theta(x))=(u_{2},\overline{\theta(x)})$, so that

[TABLE]

Kostant’s result [13, Theorem 8] then implies that $\overline{x}$ (resp. $\overline{\theta(x)}$) is the unique element of $\mathcal{S}$ with the property of being $G$-conjugate to $x$ (resp. $\theta(x)$). Since $x$ and $\theta(x)$ are themselves $G$-conjugate (by Lemma 14), it follows that $\overline{x}=\overline{\theta(x)}$. We may therefore use (22) to conclude that

[TABLE]

Now observe that $u_{1}=\pi_{U}(\psi^{-1}(x))$ and $u_{2}=\pi_{U}(\psi^{-1}(\theta(x)))$, so that

[TABLE]

and

[TABLE]

The uniqueness of $\nu(x)$ is a consequence of [15, Proposition 2.3.2]. ∎

4.3. Level sets

The map $\theta:\mathcal{V}\rightarrow\xi+\mathcal{C}$ facilitates a convenient description of Kostant’s results [15] on the level sets $\overline{F}^{-1}(z)$. To this end, let $w_{0}\in W$ denote the longest element of the Weyl group and recall the root vectors fixed in 1.2. Note that there exists a unique lift $\widetilde{w_{0}}\in N_{G}(T)$ of $w_{0}$ that satisfies

[TABLE]

for all $\alpha\in\Delta$. Now consider the open subset $U_{-}TU$ of $G$, as well as its left $\widetilde{w_{0}}$-translate

[TABLE]

The map

[TABLE]

is then a variety isomorphism. Let us invert this isomorphism and project onto the factors $U_{-}$, $T$, and $U$, thereby obtaining variety morphisms

[TABLE]

Setting $G_{x}^{*}:=G_{x}\cap G^{*}$ for all $x\in\mathfrak{g}$, one has the following rephrased version of [15, Theorem 2.6].

Theorem 16**.**

If $x\in\mathcal{V}$, then

[TABLE]

defines an isomorphism of varieties.

Now let $H(\overline{f}_{i})$ be the Hamiltonian vector field on $\mathcal{O}_{\text{KT}}$ determined by $\overline{f}_{i}$, $i\in\{1,\ldots,r\}$. Since $\overline{f}_{1},\ldots,\overline{f}_{r}$ form a completely integrable system on $\mathcal{O}_{\text{KT}}$, one knows that $H(\overline{f}_{i})$ is tangent to $\overline{F}^{-1}(\overline{F}(x))$ for all $x\in\mathcal{O}_{\text{KT}}$. It follows that for each $x\in\mathcal{V}$, $\tau_{\theta(x)}$ identifies $H(\overline{f}_{i})$ with a vector field on $G_{\theta(x)}^{*}$. The latter vector field turns out to be the restriction of the following vector field on $G_{\theta(x)}$: the left-invariant vector field corresponding to $(d_{\theta(x)}f_{i})^{\vee}\in\mathfrak{g}_{\theta(x)}$, where the notation $(d_{\theta(x)}f_{i})^{\vee}$ is as defined in 3.2. This discussion can be formulated as follows (cf. [15, Theorem 4.3]).

Theorem 17**.**

We have

[TABLE]

for all $x\in\mathcal{V}$ and $g\in G_{\theta(x)}^{*}$.

4.4. A holomorphic map $\lambda:\mathcal{V}\rightarrow G^{*}$

Let us consider the map

[TABLE]

Theorem 16 then implies that for all $x\in\mathcal{V}$, $\lambda(x)$ is the unique element of $G_{\theta(x)}^{*}$ such that

[TABLE]

The following two additional facts about $\lambda$ are needed in Section 5.

Lemma 18**.**

If $x\in\mathcal{V}$, then $\lambda(y)=\tau_{\theta(x)}^{-1}(y)$ for all $y\in\overline{F}^{-1}(\overline{F}(x))$.

Proof.

Lemma 11 implies that $x$ and $y$ are $G$-conjugate, which by Lemma 14 means that $\theta(x)=\theta(y)$. We therefore have $\lambda(y)=\tau_{\theta(y)}^{-1}(y)=\tau_{\theta(x)}^{-1}(y)$, where the first equality follows from the definition (25). ∎

Lemma 19**.**

The map $\lambda:\mathcal{V}\rightarrow G^{*}$ is holomorphic.

Proof.

Given $x\in\mathcal{V}$, we may write $\lambda(x)=\widetilde{w_{0}}u_{-}tu$ with $u_{-}\in U_{-}$, $t\in T$, and $u\in U$. Let us express $u_{-}$, $t$, and $u$ as functions of $x$. To this end, note that $u=\sigma_{U}(\lambda(x))$. Equation (26) is then $x=\operatorname{Ad}_{u}(\theta(x))$, which by Lemma 15 implies that

[TABLE]

To express $t$ in terms of $x$, we first recall that $\lambda(x)\in G_{\theta(x)}$. Hence

[TABLE]

where the last equality follows from Lemma 15. Now apply (2) and the fact that $\theta(x)\in\xi+\mathcal{C}$ to write

[TABLE]

with $x_{0}\in\mathfrak{t}$, $y\in\mathcal{C}$, and $x_{\alpha}\in\mathbb{C}^{\times}$ for all $\alpha\in\Delta$. Observe that (28) then becomes

[TABLE]

for some $z\in\mathfrak{b}_{-}$. Note that $y$ and $\operatorname{Ad}_{\widetilde{w_{0}}}(z)$ belong to $\mathfrak{b}$, while each of $\xi$ and $\sum_{\alpha\in\Delta}(\alpha(t)x_{\alpha})e_{w_{0}\alpha}$ is a sum of negative simple root vectors. It follows that

[TABLE]

or equivalently that $\alpha(t)=x_{\alpha}^{-1}$ for all $\alpha\in\Delta$. Expressing this in terms of the algebraic group isomorphism

[TABLE]

we have

[TABLE]

It remains to write $u_{-}$ in terms of $x$. Accordingly, it is not difficult to verify that

[TABLE]

We also have

[TABLE]

implying that $\operatorname{Ad}_{\widetilde{w_{0}}t}(x)\in\mathcal{V}$. An application of Lemma 14 then reveals that $\theta(\operatorname{Ad}_{\widetilde{w_{0}}t}(x))=\theta(x)$. In addition, $\widetilde{w_{0}}u_{-}\widetilde{w_{0}}^{-1}\in U$ and (28) implies that

[TABLE]

It now follows from Lemma 15 that

[TABLE]

or equivalently

[TABLE]

Our lemma follows from (27), (29), and (30), along with the fact that $\nu$ and $\gamma$ are holomorphic. ∎

5. The relationship between $\mathcal{O}_{\text{KT}}$ and $\mathcal{Z}_{\mathfrak{g}}$

We now relate the two integrable systems $\overline{F}:\mathcal{O}_{\text{KT}}\rightarrow\mathbb{C}^{r}$ and $\widetilde{F}:\mathcal{Z}_{\mathfrak{g}}\rightarrow\mathbb{C}^{r}$. Section 5.1 provides the main ingredients, namely holomorphic maps $\beta:\mathcal{V}\rightarrow\mathcal{S}$ (see (31)) and $\delta:\mathcal{V}\rightarrow U$ (see (32)) with certain desirable properties (Lemmas 20, 21, and 22). This leads to Section 5.2, which proves Theorem 1 via three propositions and two lemmas.

5.1. Some preliminaries

Recall the isomorphism (21) and let $\pi_{U}:U\times\mathcal{S}\rightarrow U$ and $\pi_{\mathcal{S}}:U\times\mathcal{S}\rightarrow\mathcal{S}$ denote the canonical projection maps. The holomorphic map

[TABLE]

then has the following important property.

Lemma 20**.**

If $x\in\mathcal{V}$, then $\beta(x)$ is the unique point in $\mathcal{S}$ with the property of being $G$-conjugate to $x$.

Proof.

We have $\psi^{-1}(x)=(u,\beta(x))$ for some $u\in U$, i.e. $\operatorname{Ad}_{u}(\beta(x))=x$. In particular, $x$ and $\beta(x)$ are $G$-conjugate. Uniqueness follows from the fact that no two distinct elements of $\mathcal{S}$ are $G$-conjugate. ∎

Now define another holomorphic map as

[TABLE]

Lemma 21**.**

If $x\in\mathcal{V}$, then $\delta(x)$ is the unique element of $U$ satisfying $\beta(x)=\operatorname{Ad}_{\delta(x)}(\theta(x))$.

Proof.

Lemmas 14 and 20 imply that $\beta(x)$ is the unique element of $\mathcal{S}$ with the property of being $G$-conjugate to $\theta(x)$. It follows that $\psi^{-1}(\theta(x))=(u,\beta(x))$ for some $u\in U$, or equivalently $\beta(x)=\operatorname{Ad}_{u^{-1}}(\theta(x))$. At the same time, (32) forces $u^{-1}=\delta(x)$ to hold. We conclude that $\beta(x)=\operatorname{Ad}_{\delta(x)}(\theta(x))$.

To establish uniqueness, suppose that $u\in U$ satisfies $\beta(x)=\operatorname{Ad}_{u}(\theta(x))$. We then have

[TABLE]

so that $\psi(u^{-1},\beta(x))=\psi(\delta(x)^{-1},\beta(x))$. Since $\psi$ is an isomorphism, we must have $u=\delta(x)$. This completes the proof. ∎

One immediate consequence of Lemma 21 is the identity

[TABLE]

The following variant of (33) is needed in the next section.

Lemma 22**.**

If $x\in\mathcal{V}$, then $G_{\beta(x)}^{*}=\delta(x)G_{\theta(x)}^{*}\delta(x)^{-1}$.

Proof.

We first establish the inclusion $\delta(x)G_{\theta(x)}^{*}\delta(x)^{-1}\subseteq G_{\beta(x)}^{*}$. Suppose that $g\in G_{\theta(x)}^{*}$ and write $g=\widetilde{w_{0}}u_{-}tu$ for some $u_{-}\in U_{-}$, $t\in T$, and $u\in U$. We have

[TABLE]

while we observe that

[TABLE]

It follows that $\delta(x)g\delta(x)^{-1}\in G^{*}$. At the same time, (33) implies that $\delta(x)g\delta(x)^{-1}\in G_{\beta(x)}$. We conclude that $\delta(x)g\delta(x)^{-1}\in G_{\beta(x)}^{*}$, which establishes the inclusion $\delta(x)G_{\theta(x)}^{*}\delta(x)^{-1}\subseteq G_{\beta(x)}^{*}$. An analogous argument gives the opposite inclusion. ∎

5.2. Proof of Theorem 1

We are now equipped to prove Theorem 1, the main result of this paper. Note that Proposition 13 shows $\mathcal{V}$ to be dense in $\mathcal{O}_{\text{KT}}$, and that Theorem 1(i) follows immediately from (20). To verify the rest of Theorem 1, recall the holomorphic maps $\lambda:\mathcal{V}\rightarrow G^{*}$, $\beta:\mathcal{V}\rightarrow\mathcal{S}$, and $\delta:\mathcal{V}\rightarrow U$ and their properties. We have $\lambda(x)\in G_{\theta(x)}^{*}\subseteq G_{\theta(x)}$ for all $x\in\mathcal{V}$, which by (33) implies that $\delta(x)\lambda(x)\delta(x)^{-1}\in G_{\beta(x)}$. It follows that $(\delta(x)\lambda(x)\delta(x)^{-1},\beta(x))\in\mathcal{Z}_{\mathfrak{g}}$, giving rise to the holomorphic map

[TABLE]

This map is compatible with the integrable systems $\overline{F}=(\overline{f}_{1},\ldots,\overline{f}_{r}):\mathcal{O}_{\text{KT}}\rightarrow\mathbb{C}^{r}$ and $\widetilde{F}=(\widetilde{f}_{1},\ldots,\widetilde{f}_{r}):\mathcal{Z}_{\mathfrak{g}}\rightarrow\mathbb{C}^{r}$ in the following sense.

Proposition 23**.**

We have a commutative diagram

[TABLE]

Proof.

Our task is to prove that \widetilde{f}_{i}\circ\varphi=\overline{f}_{i}\big{|}_{\mathcal{V}} for all $i\in\{1,\ldots,r\}$. Given $x\in\mathcal{V}$ and $i\in\{1,\ldots,r\}$, note that (10) gives

[TABLE]

Now recall that $x$ and $\beta(x)$ are $G$-conjugate (by Lemma 20). Since $f_{i}$ is $G$-invariant, it follows that $f_{i}(\beta(x))=f_{i}(x)$. This combines with the calculation (35) to show that \widetilde{f}_{i}\circ\varphi=\overline{f}_{i}\big{|}_{\mathcal{V}}, completing the proof. ∎

It remains to prove Theorem 1(iii) and show that $\varphi$ is an open embedding of complex manifolds. To address the latter issue, recall the open dense subset $\mathcal{D}\subseteq\mathbb{C}^{r}$ from 4.2. Let $\mathcal{D}_{\mathcal{S}}\subseteq\mathcal{S}$ denote the preimage of $\mathcal{D}$ under the isomorphism (9), i.e. \mathcal{D}_{\mathcal{S}}:=(F\big{|}_{\mathcal{S}})^{-1}(\mathcal{D}). It follows that $G^{*}\times\mathcal{D}_{\mathcal{S}}$ is an open subset of $G\times\mathcal{S}$, so that

[TABLE]

defines an open subset of $\mathcal{Z}_{\mathfrak{g}}$.

Lemma 24**.**

The image of $\varphi$ is $\mathcal{W}$.

Proof.

Suppose that $x\in\mathcal{V}$, recalling that $x$ and $\beta(x)$ are $G$-conjugate (see Lemma 20). This implies that $F(\beta(x))=F(x)$, while we note that (20) gives $F(x)\in\mathcal{D}$. It follows that $F(\beta(x))\in\mathcal{D}$, or equivalently \beta(x)\in(F\big{|}_{\mathcal{S}})^{-1}(\mathcal{D})=\mathcal{D}_{\mathcal{S}}. At the same time, Lemma 22 tells us that $\delta(x)\lambda(x)\delta(x)^{-1}\in G_{\beta(x)}^{*}$. We conclude that $\varphi(x)\in G^{*}\times\mathcal{D}_{\mathcal{S}}$, i.e. $\varphi(x)\in\mathcal{W}$, implying that $\varphi(\mathcal{V})\subseteq\mathcal{W}$.

To prove that $\mathcal{W}\subseteq\varphi(\mathcal{V})$, let $(g,x)\in\mathcal{W}$ be given. The restriction of $F$ to $\mathfrak{t}$ is surjective (see the proof of [13, Proposition 10]), allowing us to find $y\in\mathfrak{t}$ with $F(y)=F(x)$. Set $z:=\xi+y\in\mathcal{O}_{\text{KT}}$ and note that $F(z)=F(x)$ (see [6, Corollary 3.1.43]). Note also that $x\in\mathcal{S}\subseteq\mathfrak{g}_{\text{reg}}$, while [13, Lemma 10] implies that $z\in\mathfrak{g}_{\text{reg}}$. An application of [13, Theorem 2] then shows $z$ and $x$ to be $G$-conjugate. At the same time, the definition of $\mathcal{W}$ implies that $F(z)=F(x)\in\mathcal{D}$, so that $z\in\overline{F}^{-1}(\mathcal{D})=\mathcal{V}$. These last sentences combine with Lemma 20 to imply that $\beta(z)=x$. Lemma 22 then tells us that $G_{x}^{*}=\delta(z)G_{\theta(z)}^{*}\delta(z)^{-1}$. Now observe that $g\in G_{x}^{*}$, so that we may consider the element

[TABLE]

and its image

[TABLE]

under Kostant’s isomorphism $\tau_{\theta(z)}$ (see Theorem 16). Note that $v\in\mathcal{V}$, as $F(v)=F(z)=F(x)\in\mathcal{D}$. Note also that

[TABLE]

where the first equality comes from Lemma 18.

We claim that $\varphi(v)=(g,x)$. To this end, Lemma 11 and the condition $v\in\overline{F}^{-1}(\overline{F}(z))$ imply that $v$ and $z$ are $G$-conjugate. It then follows from Lemmas 14 and 20 that $\theta(v)=\theta(z)$ and $\beta(v)=\beta(z)$, respectively. By Lemma 21, we must have $\delta(v)=\delta(z)$. We now note that

[TABLE]

This establishes that $\mathcal{W}\subseteq\varphi(\mathcal{V})$, completing the proof. ∎

Proposition 25**.**

The map $\varphi$ is an open embedding of complex manifolds.

Proof.

As noted just before Lemma 24, $\mathcal{W}$ is an open subset of $\mathcal{Z}_{\mathfrak{g}}$. Lemma 24 therefore reduces our task to one of proving that $\varphi$ is injective. To this end, suppose that $x,y\in\mathcal{V}$ have the same images under $\varphi$, i.e.

[TABLE]

By applying Lemma 20 to the first equation, we conclude that $x$ and $y$ are $G$-conjugate. Lemma 14 then gives $\theta(x)=\theta(y)$, which by Lemma 21 implies that $\delta(x)=\delta(y)$. Since $\delta(x)\lambda(x)\delta(x)^{-1}=\delta(y)\lambda(y)\delta(y)^{-1}$, we conclude that $\lambda(x)=\lambda(y)$. The definition of $\lambda$ (see (25)) and the fact that $\theta(x)=\theta(y)$ combine to imply that $\tau_{\theta(x)}^{-1}(x)=\tau_{\theta(x)}^{-1}(y)$. This establishes that $x=y$. ∎

It remains only to prove Theorem 1(iii). To this end, recall our discussion of the vectors $(d_{x}f_{i})^{\vee}\in\mathfrak{g}_{x}$ for $x\in\mathfrak{g}$ and $i\in\{1,\ldots,r\}$ (see 3.2).

Lemma 26**.**

If $g\in G$, $x\in\mathfrak{g}$, and $y:=\operatorname{Ad}_{g}(x)$, then $\operatorname{Ad}_{g}((d_{x}f_{i})^{\vee})=(d_{y}f_{i})^{\vee}$ for all $i\in\{1,\ldots,r\}$.

Proof.

Since $f_{i}$ is a $G$-invariant polynomial on $\mathfrak{g}$, we have $f_{i}\circ\operatorname{Ad}_{g^{-1}}=f_{i}$. Differentiating both sides at $y$ yields $d_{x}f_{i}\circ\operatorname{Ad}_{g^{-1}}=d_{y}f_{i}$. It follows that for all $z\in\mathfrak{g}$,

[TABLE]

We conclude that $\operatorname{Ad}_{g}((d_{x}f_{i})^{\vee})=(d_{y}f_{i})^{\vee}$. ∎

Now observe that Theorem 1(iii) amounts to the following proposition, in which $H(\overline{f}_{i})$ (resp. $H(\widetilde{f}_{i})$) denotes the Hamiltonian vector field of $\overline{f}_{i}$ on $\mathcal{O}_{\text{KT}}$ (resp. $\widetilde{f}_{i}$ on $\mathcal{Z}_{\mathfrak{g}}$).

Proposition 27**.**

We have $H(\widetilde{f}_{i})_{\varphi(x)}=d_{x}\varphi(H(\overline{f}_{i})_{x})$ for all $i\in\{1,\ldots,r\}$ and $x\in\mathcal{V}$.

Proof.

Since $\overline{F}=(\overline{f}_{1},\ldots,\overline{f}_{r}):\mathcal{O}_{\text{KT}}\rightarrow\mathbb{C}^{r}$ is a completely integrable system, $H(\overline{f}_{i})$ is tangent to the level sets of $\overline{F}$ for each $i\in\{1,\ldots,r\}$. An application of Lemma 11 then reveals the following: if two points in $\mathcal{O}_{\text{KT}}$ lie on the same integral curve of $H(\overline{f}_{i})$, then these points are $G$-conjugate to one another. By Lemmas 14 and 20, $\theta$ and $\beta$ must be constant-valued along each integral curve of $H(\overline{f}_{i})$ in $\mathcal{V}$. Lemma 21 then implies that $\delta$ is also constant-valued along each integral curve of $H(\overline{f}_{i})$ in $\mathcal{V}$. Together with the definition (34) of $\varphi$, the previous two sentences entail the following identity for all $x\in\mathcal{V}$:

[TABLE]

where $C_{\delta(x)}:G\rightarrow G$ is conjugation by $\delta(x)$. Note that

[TABLE]

with $h:=\delta(x)\lambda(x)\delta(x)^{-1}$, and that the right hand side of (36) should be interpreted as an element of $T_{h}G\oplus\mathfrak{g}$.

Now set $g:=\lambda(x)$ in (24), noting that $\tau_{\theta(x)}(g)=x$ must also hold (see (25). We obtain

[TABLE]

or equivalently

[TABLE]

At the same time, Lemma 18 and $H(\overline{f}_{i})$ being tangent to $\overline{F}^{-1}(\overline{F}(x))$ imply that $\lambda$ and $\tau_{\theta(x)}^{-1}$ agree on any integral curve through of $H(\overline{f}_{i})$ through $x$. We conclude that

[TABLE]

so that (37) may be written as

[TABLE]

After substituting the right hand side into (36), we find that

[TABLE]

We next observe that $h:=\delta(x)\lambda(x)\delta(x)^{-1}$ satisfies $L_{h}\circ C_{\delta(x)}=C_{\delta(x)}\circ L_{\lambda(x)}$, which becomes

[TABLE]

upon the differentiation of both sides at $e\in G$. Modifying (38) accordingly, we obtain

[TABLE]

Now recall that $\operatorname{Ad}_{\delta(x)}(\theta(x))=\beta(x)$ (see Lemma 21), so that Lemma 26 yields $\operatorname{Ad}_{\delta(x)}((d_{\theta(x)}f_{i})^{\vee})=(d_{\beta(x)}f_{i})^{\vee}$. It follows that

[TABLE]

By Proposition 6, the right hand side is precisely the result of evaluating $H(\widetilde{f}_{i})$ at $\varphi(x)=(h,\beta(x))$. This completes the proof. ∎