2 Symplectic reductions

Let $M$ be a finite-dimensional complex vector space with a symplectic bilinear form $\unicode[STIX]{x1D714}$ , and let $G$ be a reductive group that acts linearly on $M$ and preserves the symplectic structure. Such an action is always Hamiltonian, i.e. it admits a moment map $\unicode[STIX]{x1D707}:M\rightarrow \mathfrak{g}^{\ast }$ , namely $\unicode[STIX]{x1D707}(m)(A)=\unicode[STIX]{x1D707}^{A}(m)=\frac{1}{2}\unicode[STIX]{x1D714}(m,Am)$ for all $m\in M$ and $A\in \mathfrak{g}$ , where $\mathfrak{g}$ denotes the Lie algebra of $G$ . Since $\unicode[STIX]{x1D707}$ is $G$ -equivariant, $G$ acts on the fibre $\unicode[STIX]{x1D707}^{-1}(\unicode[STIX]{x1D701})$ for any $\unicode[STIX]{x1D701}\in (\mathfrak{g}^{\ast })^{G}$ . The quotient $\unicode[STIX]{x1D707}^{-1}(\unicode[STIX]{x1D701})/\!\!/G$ is referred to as the symplectic reduction or Marsden–Weinstein reduction. In this paper, we are interested only in the $\unicode[STIX]{x1D701}=0$ case, which is the only choice for semisimple groups anyway, and we write $M/\!\!/\!\!/G:=\unicode[STIX]{x1D707}^{-1}(0)/\!\!/G$ . If the group $G$ is finite, the moment map vanishes identically and the symplectic reduction coincides with the ordinary quotient, $M/\!\!/\!\!/G=M/G$ .

The symplectic reduction tends to be rather singular. Despite its name, it fails in general to satisfy the criteria of a symplectic singularity in the sense of Beauville, since $M/\!\!/\!\!/G$ can be reducible or even non-reduced, as several of the examples we discuss in this paper show. However, $M/\!\!/\!\!/G$ always carries a canonical Poisson structure.

Recall that a Poisson bracket on a $\mathbb{C}$ -algebra $A$ is a $\mathbb{C}$ -bilinear Lie bracket $\{-,-\}:A\times A\rightarrow A$ that satisfies the Leibniz rule $\{fg,h\}=f\{g,h\}+g\{f,h\}$ . An ideal $I\subseteq A$ is a Poisson ideal if $\{I,A\}\subseteq I$ . Kaledin [Reference KaledinKal06] showed that all minimal primes of $A$ and the nilradical $\text{nil}(A)$ are Poisson ideals. A Poisson scheme is a scheme $X$ together with a Poisson bracket on its structure sheaf.

A symplectic form on a smooth variety induces a Poisson structure in the following way. The Hamilton vector field $H_{f}$ associated to a regular function $f$ is defined by the relation $df(\unicode[STIX]{x1D709})=\unicode[STIX]{x1D714}(H_{f},\unicode[STIX]{x1D709})$ for all tangent vectors $\unicode[STIX]{x1D709}$ . Then $\{f,f^{\prime }\}:=df(H_{f^{\prime }})=-df^{\prime }(H_{f})$ . The skew-symmetry of the bracket is immediate, and the Jacobi identity follows from $d\unicode[STIX]{x1D714}=0$ . If $X$ is a normal symplectic variety, the bracket $\{f,f^{\prime }\}$ is defined as the unique regular extension of the function $\{f|_{X_{\text{reg}}},f^{\prime }|_{X_{\text{reg}}}\}$ . For a symplectic vector space $(M,\unicode[STIX]{x1D714})$ with a basis $x_{1},\ldots ,x_{2n}$ and a dual basis $y_{1},\ldots ,y_{2n}$ characterised by $\unicode[STIX]{x1D714}(x_{i},y_{j})=\unicode[STIX]{x1D6FF}_{ij}$ , the Poisson bracket is given by $\{f,f^{\prime }\}=\sum _{i}(\unicode[STIX]{x2202}f/\unicode[STIX]{x2202}x_{i})(\unicode[STIX]{x2202}f^{\prime }/\unicode[STIX]{x2202}y_{i})$ .

The ideal $I\subseteq \mathbb{C}[M]$ of the null-fibre $\unicode[STIX]{x1D707}^{-1}(0)$ of the moment map is generated by the functions $\unicode[STIX]{x1D707}^{A}(m)=\frac{1}{2}\unicode[STIX]{x1D714}(m,Am)$ . Their differentials equal $d\unicode[STIX]{x1D707}_{m}^{A}(\unicode[STIX]{x1D709})=\unicode[STIX]{x1D714}(\unicode[STIX]{x1D709},Am)$ . Hence, for any regular function $f$ , one has $\{f,\unicode[STIX]{x1D707}^{A}\}(m)=-\unicode[STIX]{x1D714}(H_{f},Am)=-df_{m}(Am)$ . If $f$ is assumed to be $G$ -equivariant, then its derivative vanishes in the orbit directions, so that $\{f,\unicode[STIX]{x1D707}^{A}\}=0$ for all $f\in \mathbb{C}[M]^{G}$ and all $A\in \mathfrak{g}$ . The Leibniz rule now implies $\{I,\mathbb{C}[M]^{G}\}\subseteq I$ and hence $\{I^{G},\mathbb{C}[M]^{G}\}=I^{G}$ as the bracket is $G$ -invariant. This signifies that $I^{G}$ is a Poisson ideal in the invariant ring $\mathbb{C}[M]^{G}$ and, consequently, that $W/\!\!/\!\!/G=\text{Spec}((\mathbb{C}[M]/I)^{G})$ inherits a canonical Poisson structure. Note that $(\mathbb{C}[M]/I)^{G}=\mathbb{C}[M]^{G}/I^{G}$ due to the linear reductivity of $G$ .

In this paper, the relevant symplectic representations take the special form $M=V\oplus V^{\ast }$ with symplectic form $\unicode[STIX]{x1D714}(v+\unicode[STIX]{x1D711},v^{\prime }+\unicode[STIX]{x1D711}^{\prime }):=\unicode[STIX]{x1D711}(v^{\prime })-\unicode[STIX]{x1D711}^{\prime }(v)$ . Then, for any linear action of $G$ on $V$ and the corresponding contragredient action on the dual space $V^{\ast }$ , the diagonal action on $V\oplus V^{\ast }$ preserves the symplectic structure. The moment map takes the special form $\unicode[STIX]{x1D707}^{A}(v+\unicode[STIX]{x1D711})=\unicode[STIX]{x1D711}(Av)$ for $v+\unicode[STIX]{x1D711}\in V\oplus V^{\ast }$ . We refer to $V\oplus V^{\ast }$ as the symplectic double of $V$ .

3 Symplectic double of polar representations

The concept of a polar representation was developed and thoroughly investigated by Dadok and Kac. As this notion is central for our purposes, we briefly recall here the main concepts and results but refer to [Reference Dadok and KacDK85] for all proofs.

Let $V$ be a representation of a reductive group $G$ . A vector $v\in V$ is semisimple if its orbit is closed. Such orbits are affine, and by Matsushima’s criterion the stabiliser subgroup $G_{v}\subseteq G$ of $v$ is again reductive. A semisimple $v$ is regular if its orbit dimension is maximal among all semisimple orbits. For a regular $v$ consider the linear space $\mathfrak{c}_{v}:=\{x\in V\mid \mathfrak{g}x\subseteq \mathfrak{g}v\}$ . Obviously, $\mathfrak{c}_{v}=\mathfrak{c}_{w}$ for all regular semisimple elements $w\in \mathfrak{g}_{v}$ .

Dadok and Kac showed that $\mathfrak{c}_{v}$ consists of semisimple elements only and that it is annihilated by the Lie algebra $\mathfrak{g}_{v}$ of the stabiliser subgroup of $v$  [Reference Dadok and KacDK85, Lemma 2.1]. Moreover, the natural projection $\mathfrak{c}_{v}\rightarrow V/\!\!/G$ is finite [Reference Dadok and KacDK85, Proposition 2.2], so that in particular its dimension is bounded by $\dim \mathfrak{c}_{v}\leqslant \dim V/\!\!/G$ . A representation $V$ is said to be polar if there is a regular element $v$ such that $\dim \mathfrak{c}_{v}=\dim V/\!\!/G$ . The space $\mathfrak{c}_{v}$ is then called a Cartan subspace of  $V$ .

For instance, if $\dim V/\!\!/G\leqslant 1$ , then $V$ is polar and any non-zero semisimple element generates a Cartan subspace. The adjoint representation $\mathfrak{g}$ of a semisimple Lie group $G$ is polar, and any Cartan subalgebra $\mathfrak{h}\subseteq \mathfrak{g}$ is a Cartan subspace. The natural representation of $\text{SL}_{3}$ on the space $S^{3}\mathbb{C}^{3}=\mathbb{C}[x,y,z]_{3}$ of ternary cubic forms is polar, and the Hesse family $\langle x^{3}+y^{3}+z^{3},xyz\rangle$ is a Cartan subspace.

Assume now that $V$ is a polar representation of $G$ . By [Reference Dadok and KacDK85, Theorem 2.3], all Cartan subspaces are $G$ -conjugate. Let $\mathfrak{c}\subseteq V$ be a fixed Cartan subspace, and let $v\in \mathfrak{c}$ denote a regular semisimple element. Dadok and Kac further proved that there is a maximal compact Lie group $K\subseteq G$ and a $K$ -equivariant Hermitian scalar product $\langle -,-\rangle$ on $V$ such that all vectors of $\mathfrak{c}$ have minimal length in their orbit, and $\mathfrak{c}$ and $\mathfrak{g}\mathfrak{c}=\mathfrak{g}v$ are orthogonal to each other [Reference Dadok and KacDK85, Lemma 2.1]. The quotient of the normaliser subgroup $N_{G}(\mathfrak{c})=\{g\in G\mid g(\mathfrak{c})=\mathfrak{c}\}$ and the centraliser subgroup $Z_{G}(\mathfrak{c})=\{g\in G\mid g(x)=x\text{ for all }x\in \mathfrak{c}\}$ is a finite group $W=N_{G}(\mathfrak{c})/Z_{G}(\mathfrak{c})$ which, in analogy to the case of adjoint representations, is called the Weyl group of the polar representation; $W$ is generated by complex reflections when $G$ is connected. Theorem 2.10 of [Reference Dadok and KacDK85] states that the inclusion $\mathfrak{c}\rightarrow V$ induces an isomorphism

(2) $$\begin{eqnarray}\displaystyle \mathfrak{c}/W\xrightarrow[{}]{\;\cong \;}V/\!\!/G & & \displaystyle\end{eqnarray}$$

or, equivalently, $\mathbb{C}[V]^{G}\cong \mathbb{C}[\mathfrak{c}]^{W}$ . This generalises the Chevalley isomorphism in the case of adjoint representations. Since $G_{v}$ stabilises the tangent space $\mathfrak{g}v$ to the $G$ -orbit of $v$ and since the definition of $\mathfrak{c}=\mathfrak{c}_{v}$ depends only on $\mathfrak{g}v$ , the stabiliser subgroup acts on $\mathfrak{c}$ , so there are natural inclusions $Z_{G}(\mathfrak{c})\subseteq G_{v}\subseteq N_{G}(\mathfrak{c})$ of finite index. All three groups are reductive.

Our goal is to construct a morphism

(3) $$\begin{eqnarray}\displaystyle r:\mathfrak{c}\oplus \mathfrak{c}^{\ast }/W\longrightarrow V\oplus V^{\ast }/\!\!/\!\!/G & & \displaystyle\end{eqnarray}$$

and to give conditions for $r$ to be an isomorphism. As the functorial projection $V^{\ast }\rightarrow \mathfrak{c}^{\ast }$ points in the wrong direction, the first task is to identify an appropriate subspace $\mathfrak{c}^{\vee }\subseteq V^{\ast }$ that splits the projection.

Let $U\subseteq V$ be the orthogonal complement to $\mathfrak{c}\oplus \mathfrak{g}\mathfrak{c}$ . By [Reference Dadok and KacDK85, Proposition 1.3(ii)], the orthogonal decomposition

(4) $$\begin{eqnarray}\displaystyle V=\mathfrak{c}\oplus \mathfrak{g}\mathfrak{c}\oplus U & & \displaystyle\end{eqnarray}$$

is $G_{v}$ -stable, and by [Reference Dadok and KacDK85, Corollary 2.5], $\dim U/\!\!/G_{v}=0$ . As $N_{G}(\mathfrak{c})$ normalises $Z_{G}(\mathfrak{c})$ , the decomposition is also preserved by the normaliser group $N_{G}(\mathfrak{c})$ . Let

(5) $$\begin{eqnarray}\displaystyle \mathfrak{c}^{\vee }:=\{\unicode[STIX]{x1D711}\in V^{\ast }\mid \unicode[STIX]{x1D711}(\mathfrak{g}\mathfrak{c}\oplus U)=0\}. & & \displaystyle\end{eqnarray}$$

Proof. The Hermitian scalar product $\langle -,-\rangle$ defines a $K$ -equivariant semilinear isomorphism

(6) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F7}:V\rightarrow V^{\ast },\;\;v\mapsto \langle v,-\rangle . & & \displaystyle\end{eqnarray}$$

Via $\unicode[STIX]{x1D6F7}$ , $V^{\ast }$ inherits a $K$ -equivariant Hermitian scalar product $\langle \unicode[STIX]{x1D711},\unicode[STIX]{x1D713}\rangle :=\langle \unicode[STIX]{x1D6F7}^{-1}(\unicode[STIX]{x1D713}),\unicode[STIX]{x1D6F7}^{-1}(\unicode[STIX]{x1D711})\rangle$ . The orthogonality of the decomposition (4) implies that $\mathfrak{c}^{\vee }=\unicode[STIX]{x1D6F7}(\mathfrak{c})$ . Let $\mathfrak{k}$ denote the Lie algebra of $K$ . The $K$ -equivariance of the Hermitian product implies $\unicode[STIX]{x1D6F7}(\mathfrak{k}w)=\mathfrak{k}\unicode[STIX]{x1D6F7}(w)$ for any $w\in V$ . And since $\mathfrak{g}=\mathbb{C}\mathfrak{k}$ , one also has $\mathfrak{g}\unicode[STIX]{x1D6F7}(w)=\unicode[STIX]{x1D6F7}(\mathfrak{g}w)$ . In particular, for all $w\in \mathfrak{c}$ , $\langle \unicode[STIX]{x1D6F7}(w),\mathfrak{g}\unicode[STIX]{x1D6F7}(w)\rangle =0$ , so $\unicode[STIX]{x1D6F7}(w)$ is semisimple and of minimal length in its orbit by the Kempf–Ness theorem [Reference Dadok and KacDK85, Theorem 1.1]. We also deduce that $\unicode[STIX]{x1D6F7}(v)$ is regular and that $\mathfrak{c}^{\vee }=\{\unicode[STIX]{x1D711}\in V^{\ast }\mid \mathfrak{g}\unicode[STIX]{x1D711}\subseteq \mathfrak{g}\unicode[STIX]{x1D6F7}(v)\}=\mathfrak{c}_{\unicode[STIX]{x1D6F7}(v)}$ . Any element $g\in G$ may be written in the form $g=k\exp (i\unicode[STIX]{x1D709})$ with $k\in K$ and $\unicode[STIX]{x1D709}\in \mathfrak{k}$ . The $K$ -equivariance and semilinearity of $\unicode[STIX]{x1D6F7}$ yield $\unicode[STIX]{x1D6F7}(gw)=\unicode[STIX]{x1D6F7}(k\exp (i\unicode[STIX]{x1D709})w)=k\exp (-i\unicode[STIX]{x1D709})\unicode[STIX]{x1D6F7}(w)$ and hence $\unicode[STIX]{x1D6F7}(Gw)=G\unicode[STIX]{x1D6F7}(w)$ for every $w\in V$ . In particular, $\unicode[STIX]{x1D6F7}$ provides a bijection between closed orbits in $V$ and $V^{\ast }$ . Necessarily, all closed orbits in $V^{\ast }$ meet $\mathfrak{c}^{\vee }$ . This implies $\dim \mathfrak{c}^{\vee }\geqslant \dim V^{\ast }/\!\!/G$ and hence that $V^{\ast }$ is polar. The statement about the action of $W$ follows from the fact that the Weyl group can be seen as the quotient $N_{K}(\mathfrak{c})/Z_{K}(\mathfrak{c})$ according to [Reference Dadok and KacDK85, Lemma 2.7].◻

We continue to use the $K$ -equivariant semilinear automorphism $\unicode[STIX]{x1D6F7}:V\rightarrow V^{\ast }$ of (6). The proof of the proposition shows that $\unicode[STIX]{x1D6F7}(\mathfrak{c})=\mathfrak{c}^{\vee }$ and $\unicode[STIX]{x1D6F7}(\mathfrak{g}\mathfrak{c})=\mathfrak{g}\mathfrak{c}^{\vee }$ . One obtains a $K$ -equivariant orthogonal decomposition

(7) $$\begin{eqnarray}\displaystyle V^{\ast }=\mathfrak{c}^{\vee }\oplus \mathfrak{g}\mathfrak{c}^{\vee }\oplus U^{\vee } & & \displaystyle\end{eqnarray}$$

with $U^{\vee }:=\unicode[STIX]{x1D6F7}(U)$ dual to (4). As before, we consider the symplectic form $\unicode[STIX]{x1D714}(v+\unicode[STIX]{x1D711},v^{\prime }+\unicode[STIX]{x1D711}^{\prime }):=\unicode[STIX]{x1D711}(v^{\prime })-\unicode[STIX]{x1D711}^{\prime }(v)$ on $V\oplus V^{\ast }$ . The symplectic vector space $(V\oplus V^{\ast },\unicode[STIX]{x1D714})$ splits into the direct sum of symplectic subspaces $\mathfrak{c}\oplus \mathfrak{c}^{\vee }$ , $\mathfrak{g}\mathfrak{c}\oplus \mathfrak{g}\mathfrak{c}^{\vee }$ and $U\oplus U^{\vee }$ . As $\unicode[STIX]{x1D714}(\mathfrak{c}\oplus \mathfrak{c}^{\vee },\mathfrak{g}\mathfrak{c}\oplus \mathfrak{g}\mathfrak{c}^{\vee })=0$ , it follows that

(8) $$\begin{eqnarray}\displaystyle \mathfrak{c}\oplus \mathfrak{c}^{\vee }\subseteq \unicode[STIX]{x1D707}^{-1}(0), & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D707}:V\oplus V^{\ast }\rightarrow \mathfrak{g}^{\ast }$ denotes the moment map.

Polar representations behave well under taking slices [Reference Dadok and KacDK85, § 2]. Let $w\in \mathfrak{c}$ be any element. The orthogonal complement $N_{w}$ to the tangent space $\mathfrak{g}w\subseteq \mathfrak{g}\mathfrak{c}$ is stable under $G_{w}$ and contains $\mathfrak{c}$ . The representation of $G_{w}$ on $N_{w}$ is called the slice representation at $w$ , and Dadok and Kac have shown that $N_{w}$ is again polar and that $\mathfrak{c}$ is a Cartan subspace.

Proposition 3.3. Let $V$ be a polar representation of $G$ with a Cartan subspace $\mathfrak{c}\subseteq V$ and corresponding Weyl group $W$ . Let $\mathfrak{c}^{\vee }\subseteq V^{\ast }$ denote the dual Cartan subspace. Then every $m\in \mathfrak{c}\oplus \mathfrak{c}^{\vee }$ has a closed $G$ -orbit in $V\oplus V^{\ast }$ and $Gm\cap (\mathfrak{c}\oplus \mathfrak{c}^{\vee })=Wm$ . Also, the morphism

$$\begin{eqnarray}r:\mathfrak{c}\oplus \mathfrak{c}^{\vee }/W\longrightarrow V\oplus V^{\ast }/\!\!/\!\!/G\end{eqnarray}$$

is injective and finite.

Proof. The symplectic double $V\oplus V^{\ast }$ comes with a $K$ -equivariant Hermitian scalar product

$$\begin{eqnarray}\langle w+\unicode[STIX]{x1D711},w^{\prime }+\unicode[STIX]{x1D711}^{\prime }\rangle =\langle w,w^{\prime }\rangle +\langle \unicode[STIX]{x1D711},\unicode[STIX]{x1D711}^{\prime }\rangle\end{eqnarray}$$

inherited from the scalar products on its direct summands. The orthogonality of the decompositions (4) and (7) implies that $\langle m,\mathfrak{g}m\rangle =0$ for any $m\in \mathfrak{c}\oplus \mathfrak{c}^{\vee }$ . It follows from the Kempf–Ness criterion [Reference Dadok and KacDK85, Theorem 1.1] that $m$ is semisimple and of minimal length in its $G$ -orbit, and that

$$\begin{eqnarray}Gm\cap (\mathfrak{c}\oplus \mathfrak{c}^{\vee })=Km\cap (\mathfrak{c}\oplus \mathfrak{c}^{\vee }).\end{eqnarray}$$

Assume now that $(w,\unicode[STIX]{x1D711}),(w^{\prime },\unicode[STIX]{x1D711}^{\prime })\in \mathfrak{c}\oplus \mathfrak{c}^{\vee }$ belong to the same $G$ -orbit and hence the same $K$ -orbit, say $w^{\prime }=kw$ and $\unicode[STIX]{x1D711}^{\prime }=k\unicode[STIX]{x1D711}$ for some $k\in K$ . Up to the action of $W$ , we may assume $w^{\prime }=w$ , so that $k\in K_{w}=G_{w}\cap K$ . Let $x=\unicode[STIX]{x1D6F7}^{-1}(\unicode[STIX]{x1D711}),x^{\prime }=\unicode[STIX]{x1D6F7}^{-1}(\unicode[STIX]{x1D711}^{\prime })\in \mathfrak{c}$ . As $\unicode[STIX]{x1D6F7}$ is $K$ -equivariant, it follows that $kx=x^{\prime }$ . Now $x$ and $x^{\prime }$ are elements in the same $G_{w}$ -orbit and contained in a Cartan subspace $\mathfrak{c}$ of the slice representation $N_{w}$ . By [Reference Dadok and KacDK85, Theorem 2.8], $x$ and $x^{\prime }$ also belong to the same orbit under the normaliser subgroup $N_{G_{w}}(\mathfrak{c})\subseteq N_{G}(\mathfrak{c})$ . Hence, there is an element $\unicode[STIX]{x1D6FE}\in W$  with $w^{\prime }=\unicode[STIX]{x1D6FE}w=w$ and $x^{\prime }=\unicode[STIX]{x1D6FE}x$ . In particular, the natural morphism $r:\mathfrak{c}\oplus \mathfrak{c}^{\vee }/W\rightarrow V\oplus V^{\ast }/\!\!/G$ is injective. The same arguments as in the proof of [Reference Dadok and KacDK85, Proposition 2.2] show that $r$ is finite.◻

If the Cartan subspace is one-dimensional, a simple argument gives something much stronger.

The null-fibre $\unicode[STIX]{x1D707}^{-1}(0)$ of the momentum map contains the $G$ -variety

$$\begin{eqnarray}C_{0}:=\overline{G.(\mathfrak{c}\oplus \mathfrak{c}^{\vee })},\end{eqnarray}$$

which will play a key role in our analysis.

Proof. Let $J\subseteq \mathbb{C}[V\oplus V^{\ast }]$ denote the vanishing ideal of $C_{0}$ . If we can show that $\{h,f\}|_{C_{0}}=0$ for all $h\in J$ and all $f\in \mathbb{C}[V\oplus V^{\ast }]^{G}$ , then the Leibniz rule implies $\{J,\mathbb{C}[V\oplus V^{\ast }]^{G}\}\subseteq J$ , and hence $\{J^{G},\mathbb{C}[V\oplus V^{\ast }]^{G}\}\subseteq J^{G}$ , which covers the first assertion. Moreover, as $J$ is a $G$ -equivariant ideal sheaf and $G(\mathfrak{c}\oplus \mathfrak{c}^{\vee })$ is dense in $C_{0}$ by definition, it suffices to show that $\{h,f\}$ vanishes in a general point $m=w+\unicode[STIX]{x1D711}$ of $\mathfrak{c}\oplus \mathfrak{c}^{\vee }$ for all $h\in J$ and $f\in \mathbb{C}[V\oplus V^{\ast }]^{G}$ . Now the tangent space of $V\oplus V^{\ast }$ decomposes into symplectic subspaces $\mathfrak{c}\oplus \mathfrak{c}^{\vee }$ , $\mathfrak{g}\mathfrak{c}\oplus \mathfrak{g}\mathfrak{c}^{\vee }$ and $U\oplus U^{\vee }$ , and this decomposition is stable under $G_{w}$ . According to a result of Dadok and Kac mentioned above, the quotient $U/\!\!/G_{w}$ is zero-dimensional. This implies that any $G$ -invariant function $f$ is constant on subsets $m+U$ and $m+U^{\bot }$ . In particular, all derivatives of $f$ in $m$ in the directions $U\oplus U^{\vee }$ vanish. On the other hand, $h$ vanishes on $C_{0}$ so that all derivatives of $h$ vanish in the directions $\mathfrak{c}\oplus \mathfrak{c}^{\vee }$ . Thus, the calculation of the Poisson bracket is reduced to

(9) $$\begin{eqnarray}\displaystyle \{h,f\}=\mathop{\sum }_{i=1}^{2\ell }\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}x_{i}}\frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}y_{i}}, & & \displaystyle\end{eqnarray}$$

where the $x_{i}$ and the $y_{i}$ run through a basis and the dual basis of $\mathfrak{g}\mathfrak{c}\oplus \mathfrak{g}\mathfrak{c}^{\vee }$ , respectively. For a general $m$ , the tangent space $\mathfrak{g}m\subseteq \mathfrak{g}\mathfrak{c}\oplus \mathfrak{g}\mathfrak{c}^{\vee }$ is a Lagrangian subspace; as it is half-dimensional, it suffices to verify that it is isotropic. Indeed, for any $A,B\in \mathfrak{g}$ , one has

(10) $$\begin{eqnarray}\displaystyle \{Am,Bm\}=\{Aw+A\unicode[STIX]{x1D711},Bw+B\unicode[STIX]{x1D711}\}=A\unicode[STIX]{x1D711}(Bw)-B\unicode[STIX]{x1D711}(Aw)=\unicode[STIX]{x1D711}([B,A]w)=0, & & \displaystyle\end{eqnarray}$$

since $\mathfrak{c}^{\vee }$ annihilates $\mathfrak{g}\mathfrak{c}$ by construction. Let $x_{1},\ldots ,x_{\ell }$ be a basis of $\mathfrak{g}m$ and augment it by $x_{\ell +1},\ldots ,x_{2\ell }$ to form a symplectic basis of $\mathfrak{g}\mathfrak{c}\oplus \mathfrak{g}\mathfrak{c}^{\vee }$ . Then

(11) $$\begin{eqnarray}\displaystyle \{h,f\}=\mathop{\sum }_{i=1}^{\ell }\biggl(\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}x_{i}}\frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}x_{\ell +i}}-\frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}x_{i}}\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}x_{\ell +i}}\biggr). & & \displaystyle\end{eqnarray}$$

Since both functions $f$ and $h$ are constant along $Gm$ , the partial derivatives $\unicode[STIX]{x2202}f/\unicode[STIX]{x2202}x_{i}$ and $\unicode[STIX]{x2202}h/\unicode[STIX]{x2202}x_{i}$ vanish in $m$ , so that finally $\{h,f\}(m)=0$ .

For the last statement, it suffices to show that for any two functions $f,f^{\prime }\in \mathbb{C}[V\oplus V^{\ast }]^{G}$ one has

$$\begin{eqnarray}\{f,f^{\prime }\}|_{\mathfrak{ c}\oplus \mathfrak{c}^{\vee }}=\{f|_{\mathfrak{c}\oplus \mathfrak{c}^{\vee }},f^{\prime }|_{\mathfrak{ c}\oplus \mathfrak{c}^{\vee }}\},\end{eqnarray}$$

where the bracket on the left is that in $\mathbb{C}[V\oplus V^{\ast }]$ and the bracket on the right is that in $\mathbb{C}[\mathfrak{c}\oplus \mathfrak{c}^{\vee }]$ . Splitting the sum

$$\begin{eqnarray}\{f,f^{\prime }\}=\mathop{\sum }_{i}\frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}x_{i}}\frac{\unicode[STIX]{x2202}f^{\prime }}{\unicode[STIX]{x2202}y_{i}}\end{eqnarray}$$

into contributions from $\mathfrak{c}\oplus \mathfrak{c}^{\vee }$ , $\mathfrak{g}\mathfrak{c}\oplus \mathfrak{g}\mathfrak{c}^{\vee }$ and $U\oplus U^{\vee }$ , this amounts to proving that the latter two summands only contribute trivially. But this follows from the same arguments as before: the derivatives of both $f$ and $f^{\prime }$ vanish in the directions $U\oplus U^{\vee }\oplus \mathfrak{g}m$ due to their $G$ -equivariance.

The bijectivity of $r$ is obvious from Proposition 3.3 and the definition of $C_{0}$ .◻

From the two propositions above it follows that the conjecture holds if and only if the following assertions are true.

1. (i) The inclusion $C_{0}\subseteq \unicode[STIX]{x1D707}^{-1}(0)$ induces an isomorphism $C_{0}/\!\!/G\rightarrow (V\oplus V^{\ast }/\!\!/\!\!/G)_{\text{red}}$ . This is equivalent to saying that $r:\mathfrak{c}\oplus \mathfrak{c}^{\vee }/W\rightarrow V\oplus V^{\ast }/\!\!/\!\!/G$ is bijective.

2. (ii) The variety $C_{0}/\!\!/G$ is normal.

4 The structure of the null-fibre of the moment map

As before, let $V$ denote a representation of a reductive group $G$ , let $\unicode[STIX]{x1D70B}:V\rightarrow V/\!\!/G$ be the quotient map, and let $\unicode[STIX]{x1D707}:V\oplus V^{\ast }\rightarrow \mathfrak{g}^{\ast }$ be the moment map. The aim of this section is to highlight some properties of the null-fibre $\unicode[STIX]{x1D707}^{-1}(0)$ . Part of this content is directly inspired by [Reference PanyushevPan94].

Consider the linear map $f:V\times \mathfrak{g}\rightarrow V\times V$ , $(x,A)\mapsto (x,Ax)$ of trivial vector bundles on  $V$ . For each point $x\in A$ , one has $\text{rk}(f(x))=\dim \mathfrak{g}x=\dim Gx$ . A sheet $S$ of $V$ is an irreducible component of any of the strata $\{x\in V\mid \text{rk}(f(x))=r\}$ , $r\in \mathbb{N}_{0}$ , and one puts $r_{S}:=\text{rk}(f(x))$ for any $x\in S$ . The number $\text{mod}(G,S)=\dim S-r_{S}$ is called the modality of $S$ . The restriction $f|_{S}$ of $f$ to a sheet $S$ with its reduced subscheme structure has constant rank. Hence its image and its kernel are subbundles of rank $r_{S}$ and rank $\dim \mathfrak{g}-r_{S}$ , respectively. Let $\text{pr}_{1}:\unicode[STIX]{x1D707}^{-1}(0)\rightarrow V$ denote the projection to the first component of $V\oplus V^{\ast }$ . Then

$$\begin{eqnarray}\text{pr}_{1}^{-1}(S)=\{(x,\unicode[STIX]{x1D711})\in S\times V\mid \unicode[STIX]{x1D711}\bot \mathfrak{g}x=\text{Im}f(x)\}\end{eqnarray}$$

is a subbundle in $S\times V^{\ast }$ of rank $\dim V^{\ast }-r_{S}$ . In particular, it is an irreducible locally closed subset of $\unicode[STIX]{x1D707}^{-1}(0)$ of dimension

(12) $$\begin{eqnarray}\displaystyle \dim \text{pr}_{1}^{-1}(S)=\dim S+\dim V^{\ast }-r_{S}=\dim V+\text{mod}(G,S). & & \displaystyle\end{eqnarray}$$

Clearly, $\unicode[STIX]{x1D707}^{-1}(0)=\bigcup _{S}\text{pr}_{1}^{-1}(S)$ , and since the set of sheets $S$ of $V$ is finite, it follows that

$$\begin{eqnarray}\dim \unicode[STIX]{x1D707}^{-1}(0)=\dim V+\max _{S}\,\text{mod}(G,S).\end{eqnarray}$$

A representation $(G,V)$ is said to be visible if each fibre of $\unicode[STIX]{x1D70B}$ has only finitely many orbits. It is well known that it suffices to require that the special fibre $\unicode[STIX]{x1D70B}^{-1}(\unicode[STIX]{x1D70B}(0))$ have only finitely many orbits.

Let $(G,V)$ be a polar representation with Cartan subspace $\mathfrak{c}$ . The space $V$ always contains two special sheets, as described below.

Let $r^{\prime }$ be the maximal dimension of an orbit in $V$ . By semicontinuity, the set $S^{\prime }$ of points with $r^{\prime }$ -dimensional orbits is an open and hence irreducible subset of $V$ , the open sheet of $V$ . If follows that $C^{\prime }:=\overline{\text{pr}_{1}^{-1}(S^{\prime })}$ is an irreducible component of the null-fibre $\unicode[STIX]{x1D707}^{-1}(0)$ of dimension $2\dim V-r^{\prime }$ .

On the other hand, let $r_{0}$ denote the maximal dimension of a closed orbit. Let $S_{0}$ be a sheet that contains the open subset $\mathfrak{g}_{\text{reg}}\subseteq \mathfrak{c}$ of regular semisimple elements, i.e. those with $r_{0}$ -dimensional orbits. Since $\unicode[STIX]{x1D70B}:\mathfrak{c}_{\text{reg}}\rightarrow V/\!\!/G$ is dominant, so is $\unicode[STIX]{x1D70B}:S_{0}\rightarrow V/\!\!/G$ . Hence, for a general point $s\in S_{0}$ there exists a point $x\in \mathfrak{c}_{\text{reg}}$ with $\unicode[STIX]{x1D70B}(s)=\unicode[STIX]{x1D70B}(x)$ , where $\mathfrak{c}_{\text{reg}}$ denotes the dense open subset of $\mathfrak{c}$ formed by regular elements. The orbit of $x$ is closed and hence contained in the closure of the orbit of $s$ . But the two orbits have the same dimension and thus must be equal. This shows that $G\mathfrak{c}_{\text{reg}}\subseteq S_{0}\subseteq \overline{G\mathfrak{c}}$ . In particular, the sheet $S_{0}$ is uniquely determined.

In general, we have $r_{0}\leqslant r^{\prime }$ . A representation $(G,V)$ is said to be stable if the maximal orbit dimension is attained by orbits of regular semisimple elements, i.e. if $r_{0}=r^{\prime }$ .

5 Recapitulation of $\unicode[STIX]{x1D703}$ -representations

A particular class of polar representations is formed by the so-called $\unicode[STIX]{x1D703}$ -representations, as introduced by Vinberg [Reference VinbergVin76]. Let $G$ be a connected reductive group with Lie algebra $\mathfrak{g}$ . Let $\unicode[STIX]{x1D703}$ denote both an automorphism of $G$ of finite order $m$ and the induced automorphism of $\mathfrak{g}$ . After fixing a primitive $m$ th root of unity $\unicode[STIX]{x1D709}$ , the action of $\unicode[STIX]{x1D703}$ gives rise to a $\mathbb{Z}/m\mathbb{Z}$ -grading $\mathfrak{g}=\bigoplus _{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak{g}_{i}$ with components $\mathfrak{g}_{i}=\text{ker}(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D709}^{i}\,\text{id})$ . Then $\mathfrak{g}_{0}$ is a reductive Lie algebra, each $\mathfrak{g}_{i}$ is a $\mathfrak{g}_{0}$ representation, and $\mathfrak{g}_{i}$ is dual to $\mathfrak{g}_{-i}$ via the Killing form. Let $G_{0}\subseteq G$ be the connected algebraic subgroup with Lie algebra $\mathfrak{g}_{0}$ . Then $G_{0}$ is reductive and acts linearly on $\mathfrak{g}_{1}$ . The representation $(G_{0},\mathfrak{g}_{1})$ is called the $\unicode[STIX]{x1D703}$ -representation obtained from the pair $(\mathfrak{g},\unicode[STIX]{x1D703})$ . These representations are visible and polar but not always stable. A complete list of irreducible $\unicode[STIX]{x1D703}$ -representations with their main features can be found in the paper [Reference KacKac80] of Kac.

A Cartan subspace $\mathfrak{c}\subseteq \mathfrak{g}_{1}$ is defined as a maximal abelian subspace consisting of semisimple elements. The subspace $\mathfrak{c}$ lies in some $\unicode[STIX]{x1D703}$ -invariant Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$ , and the dual Cartan subspace is $\mathfrak{c}^{\vee }=\mathfrak{h}_{-1}\subseteq \mathfrak{g}_{-1}$ . The dual Cartan subspace can also be described as the space of all semisimple elements in $\mathfrak{g}_{-1}$ that commute with $\mathfrak{c}$ . Note that the moment map $\unicode[STIX]{x1D707}:\mathfrak{g}_{1}\oplus \mathfrak{g}_{-1}\rightarrow \mathfrak{g}_{0}$ is, up to the identification via the Killing form, the ordinary commutator. Hence $\unicode[STIX]{x1D707}(x,y)=0$ if and only if $x$ and $y$ commute.

If $\mathfrak{g}$ is a reductive algebra and $x\in \mathfrak{g}$ is any element, the centraliser of $x$ in $\mathfrak{g}$ will be denoted by $\mathfrak{g}^{x}=\{y\in \mathfrak{g}\mid [x,y]=0\}$ .

In [Reference PanyushevPan94, § 3.8, Theorem], Panyushev outlines the proof of the next proposition. However, he assumes the stability of the $\unicode[STIX]{x1D703}$ -representation without actually using it for the part of the proof relevant to our proposition and he considers the component $C_{0}$ instead of $\unicode[STIX]{x1D707}^{-1}(0)$ . Therefore, we have opted to reformulate his theorem in our setting and write out the details.

Proposition 5.1. Let $(G_{0},\mathfrak{g}_{1})$ be a $\unicode[STIX]{x1D703}$ -representation with Cartan subspace $\mathfrak{c}$ and let

$$\begin{eqnarray}C_{0}=\overline{G_{0}(\mathfrak{c}\oplus \mathfrak{c}^{\vee })}.\end{eqnarray}$$

Then $C_{0}/\!\!/G_{0}=\unicode[STIX]{x1D707}^{-1}(0)_{\text{red}}/\!\!/G_{0}$ .

Proof. For any $x\in \mathfrak{g}$ , denote by $x_{s}$ and $x_{n}$ its semisimple and nilpotent parts, respectively. One can check that $x_{s},x_{n}\in \mathfrak{g}_{i}$ if $x\in \mathfrak{g}_{i}$ .

Assume now that $\unicode[STIX]{x1D707}(x,y)=0$ , i.e. that $[x,y]=0$ . Then all components $x_{s}$ , $x_{n}$ , $y_{s}$ and $y_{n}$ mutually commute. According to Lemma 4.7, $(x_{s},y_{s})\in G_{0}(\mathfrak{c}\times \mathfrak{c}^{\vee })$ . Therefore, in order to prove the proposition, it suffices to show that $(x_{s},y_{s})\in \overline{G_{0}(x,y)}$ . The Lie subalgebra $\mathfrak{l}=\mathfrak{g}^{x_{s}}\cap \mathfrak{g}^{y_{s}}$ is reductive by Matsuhima’s criterion, and since $x_{s}$ and $y_{s}$ are homogeneous, $\mathfrak{l}$ is a graded subalgebra, $\mathfrak{l}=\bigoplus _{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak{l}_{i}$ . Let $L_{0}\subseteq G_{0}$ be the connected subgroup of $G_{0}$ with Lie algebra $\mathfrak{l}_{0}$ . We claim that $(0,0)\in \overline{L_{0}(x_{n},y_{n})}$ . The assertion will then follow since $\overline{L_{0}(x,y)}=(x_{s},y_{s})+\overline{L_{0}(x_{n},y_{n})}$ .

By the graded version of the Jacobson–Morozov theorem [Reference KacKac80, § 2], there exist elements $h\in \mathfrak{l}_{0}$ and $y^{\prime }\in \mathfrak{l}_{-1}$ such that $(x_{n},h,y^{\prime })$ is an $\mathfrak{s}\mathfrak{l}_{2}$ -triplet. Consider the corresponding weight space decomposition

$$\begin{eqnarray}\mathfrak{l}=\bigoplus _{j\in \mathbb{Z}}\mathfrak{l}_{h}(j),\quad \mathfrak{l}_{h}(j)=\{z\in \mathfrak{l}\mid [h,z]=jz\}\end{eqnarray}$$

and the associated one-parameter subgroup $\unicode[STIX]{x1D719}:\mathbb{G}_{m}\rightarrow L_{0}$ with $\unicode[STIX]{x1D719}(t)|_{\mathfrak{l}_{h}(j)}=t^{j}\,\text{id}_{\mathfrak{l}_{h}(j)}$ . Since $x_{n}\in \mathfrak{l}_{h}(2)$ , we have $\lim _{t\rightarrow 0}\unicode[STIX]{x1D719}(t).x_{n}=0$ . On the other hand, the centraliser of $x_{n}$ in $\mathfrak{l}$ , including $y_{n}$ , must be contained in $\bigoplus _{j\geqslant 0}\mathfrak{l}_{h}(j)$ , and so $\lim _{t\rightarrow 0}\unicode[STIX]{x1D719}(t).y_{n}=:y_{0}$ equals the weight- $0$ component of $y_{n}$ . This shows that $(0,y_{0})\in \overline{L_{0}(x_{n},y_{n})}$ . Finally, since $y_{n}$ is nilpotent, the same is true for $y_{0}$ , so that $(0,0)\in \overline{L_{0}(0,y_{0})}$ .◻

6 Slices

In the proof of our main theorem we will need an induction argument that is based on the passage from a polar representation $(G,V)$ to the slice representation $(G_{x},N)$ of a semisimple element $x\in \mathfrak{c}$ in the Cartan subspace. The reduction argument splits into two parts: a Luna-type slice theorem for symplectic reductions, and a formal Darboux theorem. We treat the slice theorem first. For the special case of a linear action of the group of units in a semisimple algebra, the theorem is due to Crawley-Boevey [Reference Crawley-BoeveyCra03, § 4]; the general case is due to Jung [Reference JungJun09]. As Jung’s paper is unpublished, we give a simplified proof as follows.

Let $(X,\unicode[STIX]{x1D714})$ be a smooth affine symplectic variety with a Hamiltonian action by a reductive group $G$ , i.e. an action that preserves the symplectic structure and admits a moment map $\unicode[STIX]{x1D707}:X\rightarrow \mathfrak{g}^{\ast }$ . Let $x\in \unicode[STIX]{x1D707}^{-1}(0)$ be a point with closed orbit and therefore reductive stabiliser subgroup $H=G_{x}$ . The defining property of the moment map, $d\unicode[STIX]{x1D707}_{x}(\unicode[STIX]{x1D709})(A)=\unicode[STIX]{x1D714}(\unicode[STIX]{x1D709},Ax)$ for all $A\in \mathfrak{g}$ and $\unicode[STIX]{x1D709}\in T_{x}X$ , directly implies that $\text{Im}(d\unicode[STIX]{x1D707}_{x})=\mathfrak{h}^{\bot }=(\mathfrak{g}/\mathfrak{h})^{\ast }$ and $\text{ker}(d\unicode[STIX]{x1D707}_{x})=(\mathfrak{g}x)^{\bot }\subseteq T_{x}X$ .

Choose a $G$ -equivariant closed embedding $X\subseteq V$ into a linear representation of $G$ and $H$ -equivariant splittings $V=\mathfrak{g}x\oplus V^{\prime }$ and $\mathfrak{g}^{\ast }=\mathfrak{h}^{\bot }\oplus \mathfrak{h}^{\ast }$ . The second splitting yields a decomposition $\unicode[STIX]{x1D707}=\unicode[STIX]{x1D707}^{\bot }\oplus \bar{\unicode[STIX]{x1D707}}$ into two components. The fact that $\text{Im}(d\unicode[STIX]{x1D707}_{x})=\mathfrak{h}^{\bot }$ implies that the component $\unicode[STIX]{x1D707}^{\bot }:X\rightarrow \mathfrak{h}^{\bot }$ is smooth at $x$ and hence $Y:=(\unicode[STIX]{x1D707}^{\bot })^{-1}(0)=\unicode[STIX]{x1D707}^{-1}(\mathfrak{h}^{\ast })$ is smooth at $x$ of dimension $\dim _{x}Y=\dim X-\dim \mathfrak{g}x$ . In fact, $T_{x}Y=\text{ker}(d\unicode[STIX]{x1D707}_{x})$ . Now $S_{X}:=(x+V^{\prime })\cap X$ and $S_{Y}:=(x+V^{\prime })\cap Y$ are transverse slices at $x$ to the orbit of $x$ in $X$ and $Y$ , respectively, and the natural projection $T_{x}S_{Y}\rightarrow (\mathfrak{g}x)^{\bot }/\mathfrak{g}x$ is an isomorphism. In particular, there is an $H$ -stable open affine neighbourhood $U$ of $x$ in $S_{Y}$ such that $U$ is smooth and the restriction of $\unicode[STIX]{x1D714}$ to $U$ is symplectic. Moreover, the composite map $\unicode[STIX]{x1D707}^{\prime }:S_{Y}\rightarrow X\xrightarrow[{}]{\;\unicode[STIX]{x1D707}\;}\mathfrak{g}^{\ast }\rightarrow \mathfrak{h}^{\ast }$ is a moment map for the action of $H$ on $S_{Y}$ . By construction, $\unicode[STIX]{x1D707}^{-1}(0)\cap S_{X}={\unicode[STIX]{x1D707}^{\prime }}^{-1}(0)$ . Hence, the inclusion induces a natural morphism

$$\begin{eqnarray}U/\!\!/\!\!/H\subseteq (\unicode[STIX]{x1D707}^{-1}(0)\cap S_{X})/\!\!/H\rightarrow \unicode[STIX]{x1D707}^{-1}(0)/\!\!/G=X/\!\!/\!\!/G,\end{eqnarray}$$

and Luna’s slice theorem implies the following result.

It remains to compare the symplectic reductions of $U$ and its tangent space $T_{x}U=\mathfrak{g}x^{\bot }/\mathfrak{g}x$ at $x$ , endowed with the linearised action of $H$ .

Let $\mathfrak{m}\subseteq \widehat{{\mathcal{O}}}_{S_{Y},x}$ denote the maximal ideal of the completion of the local ring of $S_{Y}$ at $x$ . Any choice of regular parameters $\unicode[STIX]{x1D709}_{1},\ldots ,\unicode[STIX]{x1D709}_{n}\in \mathfrak{m}$ induces an isomorphism $u:\widehat{{\mathcal{O}}}_{\mathfrak{g}x^{\bot }/\mathfrak{g}x,0}\rightarrow R$ that is characterised by $z_{i}=(\unicode[STIX]{x1D709}_{i}\,\text{mod}\,\mathfrak{m}^{2})\mapsto \unicode[STIX]{x1D709}_{i}$ , $i=1,\ldots ,n$ . Let $\unicode[STIX]{x1D714}$ denote the symplectic form on $S_{Y}$ and $\unicode[STIX]{x1D714}^{(0)}$ its value at $x$ , i.e. the corresponding constant symplectic form on $T_{x}S_{Y}$ .

Proof. Let $z_{1},\ldots ,z_{n}\in \mathfrak{m}$ be any set of regular parameters. Since $\unicode[STIX]{x1D714}$ is closed, the Poincaré lemma allows one to find $1$ -forms $\unicode[STIX]{x1D713}^{(m)}$ , $m\geqslant 2$ , with homogeneous coefficients of degree $m$ , such that $\unicode[STIX]{x1D714}$ may be written as

$$\begin{eqnarray}\unicode[STIX]{x1D714}=\mathop{\sum }_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}dz_{\unicode[STIX]{x1D6FC}}\wedge dz_{\unicode[STIX]{x1D6FD}}+d\unicode[STIX]{x1D713}^{(2)}+d\unicode[STIX]{x1D713}^{(3)}+\cdots\end{eqnarray}$$

for some non-degenerate skew-symmetric matrix $\unicode[STIX]{x1D6FA}$ . We need to find power series

$$\begin{eqnarray}\unicode[STIX]{x1D709}_{i}:=z_{i}+\mathop{\sum }_{m\geqslant 2}\unicode[STIX]{x1D709}_{i}^{(m)},\end{eqnarray}$$

where each $\unicode[STIX]{x1D709}_{i}^{(m)}$ is a homogeneous polynomial in $z_{1},\ldots ,z_{n}$ of degree $m$ , such that

(13) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}=\mathop{\sum }_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}d\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D6FC}}\wedge d\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D6FD}}. & & \displaystyle\end{eqnarray}$$

This equation is certainly satisfied if the $\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D6FD}}^{(m)}$ satisfy the relations

(14) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}^{(m)}-\mathop{\sum }_{i=1}^{m-2}\mathop{\sum }_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D6FC}}^{(m-i)}d\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D6FD}}^{(i+1)}=2\mathop{\sum }_{p,q}\unicode[STIX]{x1D6FA}_{pq}\unicode[STIX]{x1D709}_{p}^{(m)}dz_{q} & & \displaystyle\end{eqnarray}$$

for $m\geqslant 2$ . Let $\unicode[STIX]{x1D704}_{\unicode[STIX]{x2202}_{q}}$ denote the partial evaluation along the vector field $\unicode[STIX]{x2202}_{q}$ satisfying $\unicode[STIX]{x2202}_{q}(z_{p})=\unicode[STIX]{x1D6FF}_{pq}$ . Then (14) may be recast into the form

(15) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D709}_{p}^{(m)}=\frac{1}{2}\mathop{\sum }_{q}\unicode[STIX]{x1D6FA}_{pq}^{-1}\unicode[STIX]{x1D704}_{\unicode[STIX]{x2202}_{q}}\biggl(\unicode[STIX]{x1D713}^{(m)}-\mathop{\sum }_{i=1}^{m-2}\mathop{\sum }_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D6FC}}^{(m-i)}d\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D6FD}}^{(i+1)}\biggr), & & \displaystyle\end{eqnarray}$$

which provides a means of computing the $\unicode[STIX]{x1D709}^{(m)}$ , $m\geqslant 2$ , recursively.◻

Note that slice representations are defined for an arbitrary representation thanks to [Reference Popov and VinbergPV94, § 6.5] and that these coincide with slice representations of [Reference Dadok and KacDK85, § 2].

7 Proof of Theorem 1.2

We will now prove Theorem 1.2 from the introduction. Let $(G,V)$ denote a stable locally free and visible polar representation.

Panyushev [Reference PanyushevPan94, Theorem 3.2] showed that for locally free, stable, visible polar representations the null-fibre $\unicode[STIX]{x1D707}^{-1}(0)$ is a reduced and irreducible complete intersection of expected dimension $2\dim V-\dim G=\dim V+\dim \mathfrak{c}$ . In particular, the symplectic reduction is a variety, i.e. reduced and irreducible. According to Propositions 3.3 and 3.5 and Lemma 4.5, the morphism $r:\mathfrak{c}\oplus \mathfrak{c}^{\vee }/W\rightarrow V\oplus V^{\ast }/\!\!/\!\!/G$ is the normalisation of the symplectic reduction. Hence, showing that $r$ is an isomorphism is equivalent to proving that $V\oplus V^{\ast }/\!\!/\!\!/G$ is normal. (Unfortunately, it is not true in general that the null-fibre itself is normal; see Remark 5.4.)

We will argue by induction on the rank

$$\begin{eqnarray}\text{rk}(G,V)=\dim \mathfrak{c}-\dim (\mathfrak{c}\cap V^{G}).\end{eqnarray}$$

Because of the stability assumption, $\text{rk}(G,V)=0$ is equivalent to $V$ being a trivial representation. Note that we can in fact always assume that $V$ contains no trivial summand; for otherwise we may decompose $V$ as $V=\mathbb{C}^{\ell }\oplus V_{0}$ with a representation $V_{0}$ that is again stable, locally free, visible and polar. Clearly, the corresponding Cartan subspaces are related by $\mathfrak{c}=\mathbb{C}^{\ell }\oplus \mathfrak{c}_{0}$ , and the symplectic reductions by $V\oplus V^{\ast }/\!\!/\!\!/G=\mathbb{C}^{2\ell }\times (V_{0}\oplus V_{0}^{\ast })/\!\!/\!\!/G$ .

If $\text{rk}(G,V)=1$ , the assertion follows from Proposition 3.4.

Assume now that $\text{rk}(G,V)\geqslant 2$ , that $V$ contains no trivial summand, and that the assertion has been proved for all stable, locally free, visible polar representations of lower rank.

Let $(G_{w},N)$ denote the slice representation for some point $w\in \mathfrak{c}\setminus \{0\}$ . According to Lemma 6.3, the slice representation is again visible, locally free and stable. Since $w\in \mathfrak{c}\cap N^{G_{w}}$ , one has

$$\begin{eqnarray}\text{rk}(G_{w},N)=\dim \mathfrak{c}-\dim (\mathfrak{c}\cap N^{G_{w}})<\dim \mathfrak{c}=\text{rk}(G,V).\end{eqnarray}$$

By our induction hypothesis, $N\oplus N^{\ast }/\!\!/\!\!/G_{w}$ is normal. By Zariski’s theorem [Reference Zariski and SamuelZS58, Vol. II, p. 320], the completion of the local ring ${\mathcal{O}}_{N\oplus N^{\ast }/\!\!/\!\!/G_{w},[0,0]}$ is normal. By Proposition 6.5, the completion of ${\mathcal{O}}_{V\oplus V^{\ast }/\!\!/\!\!/G,[w,0]}$ is normal. Hence, the symplectic reduction is normal in a neighbourhood $B$ of the image of $(\mathfrak{c}\setminus \{0\})\times \{0\}$ and, for symmetry reasons, also in a neighbourhood $B^{\prime }$ of the image of $\{0\}\times (\mathfrak{c}^{\vee }\setminus \{0\})$ .

Consider the action of $\mathbb{C}^{\ast }\times \mathbb{C}^{\ast }$ on $V\,\oplus \,V^{\ast }$ given by $(t_{1},t_{2})\cdot (x,\unicode[STIX]{x1D711})=(t_{1}x,t_{2}\unicode[STIX]{x1D711})$ . It commutes with the $G$ -action and preserves the null-fibre. Therefore, it descends to an action on the symplectic reduction. Clearly, the orbit of any point other than the origin in the symplectic reduction meets $B$ or $B^{\prime }$ . This implies that $(V\oplus V^{\ast }/\!\!/\!\!/G)\setminus \{[0,0]\}$ is normal.

Since the rank of $(G,V)$ is at least $2$ , the codimension of the origin in the symplectic reduction is at least $4$ , so the symplectic reduction is regular in codimension 1. Using Serre’s criterion for normality, it suffices to show that the symplectic reduction satisfies condition $(S_{2})$ . According to a theorem of Crawley-Boevey [Reference Crawley-BoeveyCra03, Theorem 7.1], it suffices to verify the following assumptions in order to conclude that $V\,\oplus \,V^{\ast }/\!\!/\!\!/G$ has property $(S_{2})$ : (i)  $\unicode[STIX]{x1D707}^{-1}(0)$ has property $(S_{2})$ ; (ii)  $(V\oplus V^{\ast }/\!\!/\!\!/G)\setminus \{0\}$ has property $(S_{2})$ ; (iii)  $Z:=q^{-1}(0)\cap \unicode[STIX]{x1D707}^{-1}(0)$ has codimension at least $2$ in $\unicode[STIX]{x1D707}^{-1}(0)$ , where $q:V\oplus V^{\ast }\rightarrow V\oplus V^{\ast }/\!\!/G$ denotes the quotient map.

As we have already seen that (i) and (ii) hold, it suffices to bound the codimension of $Z:=q^{-1}(0)\cap \unicode[STIX]{x1D707}^{-1}(0)$ . Any point in $Z$ is of the form $(n,\unicode[STIX]{x1D711})$ , where $n\in V$ is nilpotent, i.e.  $0\in \overline{Gn}$ and $\unicode[STIX]{x1D711}\in (\mathfrak{g}n)^{\bot }$ . Since $V$ is visible, there are only finitely many nilpotent orbits $Gn_{1},\ldots ,Gn_{s}$ in $V$ , so $Z\subseteq \bigcup _{i=1}^{s}G.(\{n_{i}\}\times (\mathfrak{g}n_{i})^{\bot })$ and $\dim Z\leqslant \max \{\dim \mathfrak{g}n_{i}+(\dim V-\dim \mathfrak{g}n_{i})\}=\dim V$ . As $\dim \unicode[STIX]{x1D707}^{-1}(0)=\dim V+\dim \mathfrak{c}\geqslant \dim V+2$ , the theorem is proved.

8 Examples and counterexamples

We first discuss several cases where the conjecture holds but which are not covered by general results. The tools are representation theory and classical invariant theory.

Example 8.1. The conjecture holds for the representations $(\text{SL}_{n},S^{2}\mathbb{C}^{n})$ and $(\text{SL}_{n},\unicode[STIX]{x1D6EC}^{2}\mathbb{C}^{n})$ . These are polar but not $\unicode[STIX]{x1D703}$ -representations. The cases are very similar, and we will give details only for $(\text{SL}_{n},\unicode[STIX]{x1D6EC}^{2}\mathbb{C}^{n})$ with $n$ even.

Let $n=2m$ and identify both $V=\unicode[STIX]{x1D6EC}^{2}\mathbb{C}^{n}$ and its dual $V^{\ast }$ with the space of skew-symmetric matrices, so that the action of $\text{SL}_{n}$ on $V\,\oplus \,V^{\ast }$ is given by $g.(A,B)=(gAg^{t},(g^{t})^{-1}Bg^{-1})$ . Through this identification, the moment map becomes $\unicode[STIX]{x1D707}(A,B)=AB-(1/n)\,\text{tr}(AB)I_{n}$ , so that

$$\begin{eqnarray}\unicode[STIX]{x1D707}^{-1}(0)=\biggl\{(A,B)\biggl|AB=\frac{1}{n}\,\text{tr}(AB)I_{n}\biggr\}.\end{eqnarray}$$

On the other hand, we see from Schwarz’s Table 1a in [Reference SchwarzSch78] that the invariant algebra is generated by the Pfaffians $X=\text{pf}(A)$ and $Y=\text{pf}(B)$ and the traces $Z_{i}=(1/n)\,\text{tr}((AB)^{i})$ for $1\leqslant i\leqslant m-1$ . In the coordinate ring $\mathbb{C}[\unicode[STIX]{x1D707}^{-1}(0)]$ of the null-fibre, the restrictions of these invariants satisfy the relations $Z_{i}=Z_{1}^{i}$ and $XY=Z_{1}^{m}$ , and no more. It follows that $V\,\oplus \,V^{\ast }/\!\!/\!\!/\text{SL}_{n}$ is a normal surface singularity of type $A_{m-1}$ . So the conjecture holds.

Example 8.2. The conjecture holds for the standard representations $(G_{2},\mathbb{C}^{7})$ and $(F_{4},\mathbb{C}^{26})$ and the spin representation $(\text{Spin}_{7},\mathbb{C}^{8})$ . In all cases, the null-fibre of the moment map is non-reduced, but the symplectic reduction is normal. Again, the cases are very similar, and we give details only for the eight-dimensional spin representation $V$ . As $V$ has a $\text{Spin}_{7}$ -invariant non-degenerate quadratic form $q$ , the representation is self-dual, $V\cong V^{\ast }$ . According to Schwarz [Reference SchwarzSch07, Theorem 4.3], the invariant algebra $\mathbb{C}[V\oplus V]^{\text{Spin}_{7}}$ is generated by the invariants $A(v+v^{\prime })=q(v)$ and $B(v+v^{\prime })=q(v,v^{\prime })$ , the polarisation of $v$ , and $C(v+v^{\prime })=q(v^{\prime })$ . The quadratic part of the coordinate ring $\mathbb{C}[V\oplus V]$ equals $S^{2}V\oplus V\otimes V\oplus S^{2}V$ , and the summand $V\otimes V$ that contains the generators of the ideal $I$ of the null-fibre $\unicode[STIX]{x1D707}^{-1}(0)$ further decomposes into

$$\begin{eqnarray}V\otimes V=S^{2}V\oplus \unicode[STIX]{x1D6EC}^{2}V\cong (V_{(0,0,0)}\oplus V_{(0,0,2)})\oplus (V_{(0,1,0)}\oplus V_{(1,0,0)}),\end{eqnarray}$$

where $V_{(k_{1},k_{2},k_{3})}$ denotes the irreducible representation with highest weight $k_{1}\unicode[STIX]{x1D71B}_{1}+k_{2}\unicode[STIX]{x1D71B}_{2}+k_{3}\unicode[STIX]{x1D71B}_{3}$ . One can check that $I$ is in fact generated by $V_{(0,1,0)}$ , and that $J^{2}\subseteq I\subsetneq J$ , where $J$ is the ideal generated by $V_{(0,1,0)}\oplus V_{(0,0,2)}$ . In particular, $\unicode[STIX]{x1D707}^{-1}(0)$ is non-reduced. A direct computation with Singular [Reference Decker, Greuel, Pfister and SchönemannDGPS15] or Macaulay2 [Reference Grayson and StillmanGS] shows that $\mathbb{C}[\unicode[STIX]{x1D707}^{-1}(0)]^{\text{Spin}_{7}}=\mathbb{C}[A,B,C]/(AC-B^{2})$ is a normal ring.

Example 8.3. The conjecture holds for the spin representation $(\text{Spin}_{9},V=\mathbb{C}^{16})$ . In this example, even the symplectic reduction is non-reduced, but $(V\oplus V^{\ast }/\!\!/\!\!/\text{Spin}_{9})_{\text{red}}$ is normal. As before, $V_{(k_{1},k_{2},k_{3},k_{4})}$ will denote the irreducible representation of highest weight $k_{1}\unicode[STIX]{x1D71B}_{1}+k_{2}\unicode[STIX]{x1D71B}_{2}+k_{3}\unicode[STIX]{x1D71B}_{3}+k_{4}\unicode[STIX]{x1D71B}_{4}$ . In particular, the spin representation is $V=V_{(0,0,0,1)}$ . As for $\text{Spin}_{7}$ , there is an invariant quadratic form $q$ on $V$ that allows the identification of $V$ and $V^{\ast }$ , and whose polarisations provide invariants of bidegree $(2,0)$ , $(1,1)$ and $(0,2)$ in $\mathbb{C}[V\oplus V]^{\text{Spin}_{9}}$ . It is well known that the null-fibre $\unicode[STIX]{x1D707}^{-1}(0)$ has two irreducible components (cf. [Reference PanyushevPan04, § 4]). Let $I\subseteq \mathbb{C}[V\oplus V]$ denote the vanishing ideal of the null-fibre. It is generated by the 36-dimensional summand $V_{(0,1,0,0)}(\cong \mathfrak{s}\mathfrak{o}_{9})$ in the decomposition

$$\begin{eqnarray}\mathbb{C}[V\oplus V]_{2}\supseteq V\otimes V\cong (V_{(0,0,0,0)}\oplus V_{(0,0,0,2)}\oplus V_{(1,0,0,0)})\oplus (V_{(0,0,1,0)}\oplus V_{0,1,0,0}).\end{eqnarray}$$

Moreover, according to Schwarz [Reference SchwarzSch78, Table 3a], the invariant ring $\mathbb{C}[V\oplus V]^{\text{Spin}_{9}}$ is generated by the aforementioned invariants $A$ , $B$ and $C$ together with an invariant $D$ of bidegree $(2,2)$ . The explicit description of $D$ is more involved. It arises as the unique invariant quadratic form on $V_{0,1,0,0}\cong \unicode[STIX]{x1D6EC}^{3}\mathbb{C}^{9}$ .

Using Singular or Macaulay2, one can check that the invariant $P=AC-B^{2}$ satisfies $P\not \in I$ but $P^{2}\in I$ , so that $V\oplus V^{\ast }/\!\!/\!\!/\text{Spin}_{9}$ is not reduced.

From Tevelev’s theorem [Reference TevelevTev00] we already know that the conjecture holds for the spin representation of $\text{Spin}_{9}$ , since it can be realized as the isotropy representation of the symmetric space $F_{4}/\text{Spin}_{9}$ , i.e. a $\unicode[STIX]{x1D703}$ -representation of order $m=2$ . Explicitly, this can be seen as follows. First, one verifies (for instance by using the software LiE [Reference van Leeuwen, Cohen and LisservLCL92]) that $V\otimes V\otimes V_{(0,1,0,0)}$ contains a unique copy of the trivial representation, and thus the bidegree- $(2,2)$ component $I_{(2,2)}$ of the ideal $I$ contains at most one invariant. Using Macaulay2, one computes the rank of the multiplication map $\unicode[STIX]{x1D705}:I_{(1,1)}\otimes I_{(1,1)}\rightarrow I_{(2,2)}$ to be 666. The only dimension match for a submodule in $V\otimes V\otimes V_{(0,1,0,0)}$ is $\text{Im}(\unicode[STIX]{x1D705})=V_{(0,0,0,0)}\oplus V_{(0,0,0,2)}\oplus V_{(0,2,0,0)}\oplus V_{(2,0,0,0)}$ . In particular, $I_{(2,2)}^{\text{Spin}_{9}}=\mathbb{C}Q$ for some linear combination $Q=\unicode[STIX]{x1D6FC}D\,\oplus \,\unicode[STIX]{x1D6FD}AC+\unicode[STIX]{x1D6FE}B^{2}$ with $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D6FE}\in \mathbb{C}$ . A Macaulay2 calculation gives $I\cap \mathbb{C}[A,B,C]=(A,B,C)(AC-B^{2})$ . Hence, the coefficient $\unicode[STIX]{x1D6FC}$ must be non-zero. It follows that $I^{\text{Spin}_{9}}=(Q,A(AC-B^{2}),B(AC-B^{2}),C(AC-B^{2}))$ and $\sqrt{I^{\text{Spin}_{9}}}=(Q,AC-B^{2})$ , whence

$$\begin{eqnarray}\mathbb{C}[\unicode[STIX]{x1D707}^{-1}(0)_{\text{red}}]^{\text{Spin}_{9}}=\mathbb{C}[A,B,C,D]/(Q,AC-B^{2})=\mathbb{C}[A,B,C]/(AC-B^{2}),\end{eqnarray}$$

a normal ring.

Remark 8.4. One of the features of the representation $\text{Spin}_{9}$ is that $\unicode[STIX]{x1D707}^{-1}(0)$ is reducible. Hence, one might ask whether for every isotropy representation $(G,V)$ of a symmetric space whose commuting scheme $\unicode[STIX]{x1D707}^{-1}(0)$ is reducible we always have that $V\oplus V^{\ast }/\!\!/\!\!/G$ is non-reduced. The answer is actually negative. For instance, if

$$\begin{eqnarray}(G,V)=(\text{GL}(V_{1})\times \text{GL}(V_{2}),\,\text{Hom}(V_{1},V_{2})\times \text{Hom}(V_{2},V_{1}))\end{eqnarray}$$

with $\dim V_{1}=1<\dim V_{2}$ , then $(G,V)$ is the isotropy representation of a symmetric space [Reference Panyushev and YakimovaPY07, Example 4.3], $\unicode[STIX]{x1D707}^{-1}(0)$ is reduced and has three irreducible components [Reference TerpereauTer12, § 2.2.1], and $V\oplus V^{\ast }/\!\!/\!\!/G$ is a normal variety [Reference TerpereauTer12, § 2.2.2].

The following proposition indicates that the visibility assumption in Conjecture 1.1 should not be dropped.