2.1 Notation

Throughout, $\mathbb {N} := \{ 0,1,2,\dots \} = \mathbb {Z}_{\ge 0}$ .

2.2 Graphs

We fix a finite unoriented graph $\mathsf {G}$ with vertex set $I = \{ 0,1, \dots , r \}$ . Let H be the set of pairs consisting of an edge together with a choice of orientation of that edge. For $h \in H$ , write $s(h)$ for the source vertex of h and $t(h)$ for the target vertex. To this graph we associate a root lattice $Q = \bigoplus _{i \in I} \mathbb {Z} e_i$ and a weight lattice $P = \bigoplus _{i \in I} \mathbb {Z} \Lambda _i$ . We set

$$\begin{align*}P^+ = \sum_{i = 0}^r \mathbb{Z}_{\ge 0} \Lambda_i, \quad Q^+ = \sum_{i = 0}^r \mathbb{Z}_{\ge 0} e_i. \end{align*}$$

For $\alpha \in Q$ , we write throughout $\alpha = \sum _{i \in I} \alpha _i e_i$ , where $\alpha _i \in \mathbb {Z}$ . If $a_{i,j}$ denotes the number of edges between vertices $i,j \in I$ then we define a symmetric bilinear form $( - , - )$ on Q, called the Cartan pairing, by

$$\begin{align*}(e_i,e_j) = 2 \cdot 1_{i,j} - a_{i,j}, \end{align*}$$

where $1_{i,j}$ is the Kronecker delta. If i is a loopfree vertex then there is a reflection $s_i \colon Q \to Q$ defined by $s_i(\alpha ) = \alpha - (\alpha , e_i) e_i$ .

There is also the dual reflection,

$$\begin{align*}s_i^* \colon \operatorname{{\mathrm{Hom}}}_{\mathbb{Z}}(Q,\mathbb{C}) \to \operatorname{{\mathrm{Hom}}}_{\mathbb{Z}}(Q,\mathbb{C}). \end{align*}$$

For an element $\mathbf {c}\in \operatorname {{\mathrm {Hom}}}_{\mathbb {Z}}(Q,\mathbb {C})$ , we abbreviate $\mathbf {c}_i=\mathbf {c}(e_i)$ . In particular, we identify $\operatorname {{\mathrm {Hom}}}_{\mathbb {Z}}(Q,\mathbb {C})$ with ${\mathbb {C}}^I$ . The dual reflection $s_i^*$ is given by $(s_i^* \mathbf {c})_j = \mathbf {c}_j - (e_i,e_j)\mathbf {c}_i$ . The subgroup $W_{\mathsf {G}}$ of the group of automorphisms of the abelian group Q generated by all reflections $s_i$ is the Weyl group associated to the graph. For each $\mathbf {w}=\sum _{i\in I}\mathbf {w}_i\Lambda _i \in P$ , there is also a twisted action of $W_{\mathsf {G}}$ on Q given by

(2.1) $$ \begin{align} s_i \star_{\mathbf{w}} \alpha = s_i(\alpha) + \mathbf{w}_i e_i. \end{align} $$

Remark 2.2. When the graph contains no loops, one can associate to $\mathsf {G}$ a Kac–Moody algebra $\mathfrak {g}$ [Reference Kac24] with Cartan subalgebra $\mathfrak {h}$ . Then the $e_i$ are the simple roots and the $\Lambda _i$ are the fundamental weights, so that Q is the root lattice, and P is the lattice of weights of the derived subalgebra $\mathfrak {g}':=[\mathfrak {g},\mathfrak {g}]$ . As a result we have a natural inclusion $Q \subset \mathfrak {h}^*$ , and if we make a choice of complement $\mathfrak {d}$ of $\mathfrak {h} \cap \mathfrak {g}'$ in $\mathfrak {h}$ , this induces another natural inclusion $P \subset \mathfrak {d}^\perp \subset \mathfrak {h}^*$ , with $P+Q$ a full lattice in $\mathfrak {h}^*$ . Moreover, the (possibly degenerate) Cartan pairing on Q extends to a nondegenerate pairing on $\mathfrak {h}^*$ such that $(\Lambda _i, e_j)=\delta _{ij}$ , and the action of $W_{\mathsf {G}}$ extends to $\mathfrak {h}^*$ , defined by

$$\begin{align*}s_i (\mathbf{w}) = \mathbf{w} - (\mathbf{w}, e_i) e_i=\mathbf{w} - \mathbf{w}_i e_i. \end{align*}$$

Then $s_i(\mathbf {w} - \alpha ) = \mathbf {w} - s_i \star _{\mathbf {w}} \alpha $ . Alternatively, the action $\star _{\mathbf {w}}$ can be viewed as coming from the usual action on dimension vectors of the Weyl group of the deframed graph; see Section 2.5 and [Reference Nakajima37, Definition 2.3].

In the case of an affine Dynkin graph with $0$ a chosen extending vertex, the subspace $\mathfrak {d}$ is a line, and we choose it to be annihilated by all of the simple roots $e_i, i>0$ of the finite Dynkin subgraph. The resulting lift of P to $\mathfrak {h}^*$ is independent of the choice of $\mathfrak {d}$ up to an automorphism of $\mathfrak {g}$ satisfying $d \mapsto d+tc$ for $d \in \mathfrak {d}$ and c a central element of $\mathfrak {g}$ .

2.3 Root systems

Set $p(\alpha ) = 1 - (1/2) (\alpha , \alpha )$ . The support of a vector $\alpha \in Q$ is the full subgraph whose vertices are $\{ i \in I \, | \, \alpha _i \neq 0 \}$ . The fundamental region F is the set of all $\alpha \in Q^+$ with connected support and with $(\alpha , e_i) \le 0$ for all i. The real roots (respectively imaginary roots) are the elements of Q which can be obtained from the coordinate vector at a loopfree vertex (respectively $\pm $ an element of the fundamental region) by applying some sequence of reflection at loopfree vertices. Recall that a root $\beta $ is isotropic imaginary if $p(\beta ) = 1$ (equivalently $(\beta , \beta ) = 0$ ) and anisotropic imaginary if $p(\beta )> 1$ . Abusing terminology, we will simply say that a root $\alpha $ is (a) real if $p(\alpha ) = 0$ , (b) isotropic if $p(\alpha ) = 1$ , and (c) anisotropic if $p(\alpha )> 1$ .

We recall the following important set, first defined by Crawley-Boevey. Fix $\mathbf {c} \in \operatorname {{\mathrm {Hom}}}(Q,\mathbb {C})$ . Let $\Sigma _{\mathbf {c}}$ denote the set of all positive roots $\alpha $ such that $\mathbf {c}(\alpha ) = 0$ and

$$\begin{align*}p(\alpha)> p(\beta^{(0)}) + \cdots + p(\beta^{(k)}) \end{align*}$$

for all proper decompositions $\alpha = \beta ^{(0)} + \cdots + \beta ^{(k)}$ into a sum of positive roots $\beta ^{(i)}$ also satisfying $\mathbf {c}(\beta ^{(i)}) = 0$ . Then [Reference Crawley-Boevey15, Theorem 1.2] says that there exists a simple representation of dimension $\alpha $ for the deformed preprojective algebra $\Pi ^{\mathbf {c}}$ associated to $\mathsf {G}$ if and only if $\alpha \in \Sigma _{\mathbf {c}}$ .

2.4 Quiver varieties

We fix a pair $(\mathbf {w},\alpha ) \in P^+ \times Q^+$ . Let $(V_i)_{i \in I}$ and $(W_i)_{i \in I}$ be tuples of complex vector spaces with $\dim V_i = \alpha _i$ , $\dim W_i = \mathbf {w}_i$ . Consider the action of the group $G(\alpha ) := \prod _{i \in I} GL(\alpha _i)$ on the space

(2.2) $$ \begin{align} \operatorname{{\mathrm{Rep}}}(\mathbf{w},\alpha) := \bigoplus_{h \in H} \operatorname{{\mathrm{Hom}}}_{\mathbb{C}}(V_{s(h)}, V_{t(h)}) \bigoplus_{i \in I} \operatorname{{\mathrm{Hom}}}_{\mathbb{C}}(W_i,V_i) \bigoplus_{j \in I} \operatorname{{\mathrm{Hom}}}_{\mathbb{C}}(V_j,W_j). \end{align} $$

There is an involution $(-)^*$ on H such that $h^*$ has the same underlying edge as h but opposite orientation. Fix $\epsilon \colon H \to \{ \pm 1 \}$ such that $\epsilon (h^*) = - \epsilon (h)$ . The subset of oriented edges (arrows) h in H with $\epsilon (h) = 1$ form a quiver ${\mathsf {Q}}$ whose double $\overline {{\mathsf {Q}}}$ has arrow set H. The function $\epsilon $ specifies a symplectic form on $\operatorname {{\mathrm {Rep}}}(\mathbf {w},\alpha )$ , making the action of $G(\alpha )$ Hamiltonian. We identify $\mathfrak {g}(\alpha )$ with its dual using the trace form so that the moment map $\mu \colon \operatorname {{\mathrm {Rep}}}(\mathbf {w},\alpha ) \to \mathfrak {g}(\alpha )$ for this action, uniquely specified by $\mu (0) = 0$ , is given by

$$ \begin{align*}\mu(X,v,w) = \sum_{h \in H} \epsilon(h) X_{h^*} X_h + w v. \end{align*} $$

We will sometimes write $\mu _\alpha $ instead of $\mu $ to specify $\alpha $ . Let $\mathbf {c} \in \operatorname {{\mathrm {Hom}}}_{\mathbb {Z}}(Q,\mathbb {C}) \cong \mathbb {C}^I$ , identified with the tuple of scalar matrices $(\mathbf {c}_i \operatorname {{\mathrm {Id}}}_{V_i})_{i \in I} \in \mathfrak {g}(\alpha )$ . The framed quiver variety is

$$\begin{align*}\mathfrak{M}_{\mathbf{c}}(\mathbf{w},\alpha) = \mu^{-1}(\mathbf{c}) /\!\!/\! \, G(\alpha). \end{align*}$$

In the above, we equip $\mathfrak {M}_{\mathbf {c}}(\mathbf {w},\alpha )$ with the reduced scheme structure. Then $\mathfrak {M}_{\mathbf {c}}(\mathbf {w},\alpha )$ is an irreducible normal affine variety [Reference Crawley-Boevey15, Reference Crawley-Boevey16, Reference Crawley-Boevey17]. Except for sections 8 and 9, we will be exclusively interested in the case where $\mathbf {w} = \Lambda _0$ . In this case, we write

$$\begin{align*}{\mathcal{X}}_{\mathbf{c}}(\alpha) := \mathfrak{M}_{\mathbf{c}}(\Lambda_0,\alpha) \end{align*}$$

for brevity. When $\mathbf {w} = 0$ , we write $\mathfrak {M}_{\mathbf {c}}(\alpha ) = \mathfrak {M}_{\mathbf {c}}(0,\alpha )$ and $\operatorname {{\mathrm {Rep}}}(0,\alpha )=\operatorname {{\mathrm {Rep}}}(\alpha )$ . We may also sometimes write $\operatorname {{\mathrm {Rep}}}(\overline {\mathsf {Q}},\alpha )$ instead of $\operatorname {{\mathrm {Rep}}}(\alpha )$ to recall that this is the representation space for the double quiver $\overline {\mathsf {Q}}$ . The definition of $\mathfrak {M}_{\mathbf {c}}(\mathbf {w},\alpha )$ depends on the choice of function $\epsilon $ (equivalently, on the quiver ${\mathsf {Q}}$ ), though there is a canonical isomorphism between the spaces $\mathfrak {M}_{\mathbf {c}}(\mathbf {w},\alpha )$ defined using different $\epsilon $ . When we need to indicate the dependence of $\mathfrak {M}_{\mathbf {c}}(\mathbf {w},\alpha )$ on ${\mathsf {Q}}$ we write $\mathfrak {M}_{\mathbf {c}}({\mathsf {Q}},\mathbf {w},\alpha )$ and when we only need to specify $\mathsf {G}$ , we’ll write $\mathfrak {M}_{\mathbf {c}}(\mathsf {G},\mathbf {w},\alpha )$ .

2.5 Deframing

If $\mathbf {w} \neq 0$ , one can always use Crawley-Boevey’s trick to “deframe” the framed quiver and realize $\mathfrak {M}_{\mathbf {c}}(\mathbf {w},\alpha )$ as an unframed quiver variety $\mathfrak {M}_{{\mathbf {c}}_{\alpha }}(e_{\infty } + \alpha )$ associated to a new graph $\mathsf {G}'$ where the new vertex set $I'$ has one additional vertex ${\infty }$ and there are $\mathbf {w}_i$ additional edges between the vertex ${\infty }$ and vertex i and no loops at $\infty $ . Here ${\mathbf {c}}_{\alpha }(e_i) = \mathbf {c}(e_i)$ for $i \neq \infty $ and ${\mathbf {c}}_{\alpha }(e_{\infty }) = - \mathbf {c} (\alpha )$ . Moreover, we have defined the action $\star _{\mathbf {w}}$ in (2.1) precisely so that $s_i(e_{\infty } + \beta ) = e_{\infty } + s_i \star _{\mathbf {w}} \beta $ for $i \in I \subset I'$ and $\beta \in Q \subset Q'$ . We define $E^{\mathbf {w}}_{\mathbf {c}} \subset Q^+$ to be the set of dimension vectors $\gamma \in Q^+$ such that $e_{\infty } + \gamma \in \Sigma _{{\mathbf {c}}_{\gamma }}(\mathsf {G}')$ .

When the graph $\mathsf {G}$ has no loops, we can associate to it, as in Remark 2.2, a Kac–Moody Lie algebra $\mathfrak {g}$ with Cartan subalgebra $\mathfrak {h}$ such that $Q \subset \mathfrak {h}^*$ is the root lattice of $\mathfrak {g}$ , and P is the lattice of weights of the derived subalgebra $\mathfrak {g}'=[\mathfrak {g},\mathfrak {g}]$ . We can lift P to a subset of the dual $\mathfrak {h}^*$ , and then $P+Q$ spans $\mathfrak {h}^*$ . Thus sums in $P+Q$ can be interpreted as weights of $\mathfrak {g}$ .

With this in mind, the dimension vector $\mathbf {w} \in P^+$ labels an integrable highest weight $\mathfrak {g}$ -module $L(\mathbf {w})$ . For a fixed $\mathbf {w} \in P^+$ , we define the quadratic function

$$\begin{align*}p_{\mathbf{w}}(\beta) := p(\beta) + {\mathbf{w} \cdot \beta} - 1, \quad \forall \, \beta \in Q. \end{align*}$$

Then $p_{\mathbf {w}}(\beta ) = p(e_{\infty } + \beta )$ . It follows from [Reference Nakajima38, Theorem 2.15] that we can equivalently define $E_{\mathbf {c}}^{\mathbf {w}}$ to be set of $\gamma \in Q^+$ such that

1. (i) $\mathbf {w} - \gamma $ is a weight of $L(\mathbf {w})$ ;

2. (ii) $p_{\mathbf {w}}(\gamma )> p_{\mathbf {w}}(\beta ^{(0)}) + p(\beta ^{(1)}) + \cdots + p(\beta ^{(k)})$ , where $\gamma = \beta ^{(0)} + \beta ^{(1)} + \cdots + \beta ^{(k)}$ is a proper decomposition with $\beta ^{(0)} \in Q^+$ , $\mathbf {w} - \beta ^{(0)}$ a weight of $L(\mathbf {w})$ and $\beta ^{(i)} \in \Sigma _{\mathbf {c}}(\mathsf {G})$ for $i \ge 1$ .

This is because $e_{\infty } + \gamma $ is a root for $\mathsf {G}'$ if and only if $\mathbf {w} - \gamma $ is a weight of the representation $L(\mathbf {w})$ . We note that $0 \in E_{\mathbf {c}}^{\mathbf {w}}$ but $0$ never belongs to $\Sigma _{\mathbf {c}}$ .

When $\mathbf {w} = \Lambda _0$ , we drop $\mathbf {w}$ from the notation and write $E_{\mathbf {c}}=E_{\mathbf {c}}^{\Lambda _0}$ .

2.6 Admissible reflections

A reflection $s_i$ is said to be $\mathbf {c}$ -admissible if $\mathbf {c}_i \neq 0$ . In this case, it was shown by Maffei [Reference Maffei29] (and Nakajima [Reference Nakajima37] in the hyperkähler setting) that there is an isomorphism $\mathfrak {M}_{\mathbf {c}}(\mathbf {w},\alpha ) \cong \mathfrak {M}_{s_i^*(\mathbf {c})}(\mathbf {w},s_i \star _{\mathbf {w}} \alpha )$ , where $\mathfrak {M}_{s_i^*(\mathbf {c})}(\mathbf {w},s_i \star _{\mathbf {w}} \alpha ) = \emptyset $ if $s_i \star _{\mathbf {w}} \alpha \notin Q^+$ . Moreover, by [Reference Losev28, Lemma 6.4.3] this isomorphism is Poisson.

More generally, we say that $w \in W_{\mathsf {G}}$ is $\mathbf {c}$ -admissible if there exists a decomposition $w = s_{i_k} \cdots s_{i_1}$ such that $s_{i_j}$ is $s_{i_{j-1}}^* \cdots s_{i_1}^*(\mathbf {c})$ -admissible. In this case, $\mathfrak {M}_{\mathbf {c}}(\mathbf {w},\alpha ) \cong \mathfrak {M}_{w^*(\mathbf {c})}(\mathbf {w},w \star _{\mathbf {w}} \alpha )$ .

If $\mathbf {c}' \in W_{\mathsf {G}} \cdot \mathbf {c}$ and $w \in W_{\mathsf {G}}$ an element of minimal length with the property that $w^*(\mathbf {c}) = \mathbf {c}'$ then [Reference Crawley-Boevey and Holland18, Corollary 5.2] says that w is $\mathbf {c}$ -admissible.

2.7 The symplectic leaves

The quiver variety $\mathfrak {M}_{\mathbf {c}}(\mathbf {w},\alpha )$ has a finite stratification by (locally closed, smooth) symplectic leaves. This stratification equals the stratification by representation type [Reference Bellamy and Schedler4, Theorem 1.9]. To explain what this means, we first recall that a decomposition $\tau $ of $\alpha $ with respect to $(\mathbf {w},\mathbf {c})$ is tuple

(2.3) $$ \begin{align} \tau = (\beta^{(0)},(n_1,\beta^{(1)}; \dots;n_k,\beta^{(k)})), \end{align} $$

where $\beta ^{(0)} \in E^{\mathbf {w}}_{\mathbf {c}}$ , $\beta ^{(i)} \in \Sigma _{\mathbf {c}}$ for $i \ge 1$ and $\alpha = \beta ^{(0)} + \sum _{i = 1}^k n_i \beta ^{(i)}$ . Note that the $\beta ^{(i)}$ need not be pairwise distinct. In the case $\mathbf {w} = 0$ , the term $\beta ^{(0)}$ is omitted and we think of $\tau $ as a function $\tau \colon \Sigma _{\mathbf {c}} \to \mathcal {P}$ such that $\sum _{\beta \in \Sigma _{\mathbf {c}}} |\tau (\beta )|\beta = \alpha $ . Here $\tau (\beta ) := (n_{i_1} \ge n_{i_2} \ge \dots )$ if $i_1,i_2,\dots $ are all indices such that $\beta ^{(i_j)} = \beta $ .

The leaf ${\mathfrak L}_{\tau }$ labeled by the decomposition $\tau $ parameterizes all isomorphism classes $[M]$ of representations $M = M_0 \oplus \left (\bigoplus _{i = 1}^k M_i^{\oplus n_i} \right )$ , where each $M_i$ for $i\ne 0$ is a simple representation of the deformed preprojective algebra $\Pi ^{\mathbf {c}_{\alpha }}(\mathsf {G}')$ of dimension $\beta ^{(i)}$ , $M_0$ is the representation of dimension $e_\infty +\beta ^{(0)}$ , and $M_i \not \cong M_j$ for $i \neq j$ . If a real root occurs more than once in $\tau $ then ${\mathfrak L}_{\tau } = \emptyset $ . Otherwise, it is nonempty. The result [Reference Crawley-Boevey15, Theorem 1.3] says that when $\tau $ is written in the form (2.3) then

(2.4) $$ \begin{align} \dim {{\mathfrak L}}_{\tau} = 2 \left( p_{\mathbf{w}}(\beta^{(0)}) + \sum_{i = 1}^{k} p(\beta^{(i)}) \right). \end{align} $$

We note that the stratification by symplectic leaves satisfies the frontier condition: ${\mathfrak L}\ \cap\ \overline {\mathcal {M}} \neq \emptyset $ implies that ${\mathfrak L} \subset \overline {\mathcal {M}}$ .

2.8 Leaf closures

In Sections 2.8 and 2.9, we assume $\mathbf {w} = 0$ . Note that, as explained in Section 2.5, we can always deframe a framed quiver variety. Therefore, our results in 2.8 and 2.9 also apply to framed quiver varieties. Recall that for a decomposition $\tau $ , ${\mathfrak L}_{\tau }$ is the symplectic leaf labeled by $\tau $ .

We give the proof of Theorem 1.9, which we recall below.

Theorem 2.3. There is a morphism $\prod _{\beta \in \Sigma _{\mathbf {c}}} \mathfrak {M}_{\mathbf {c}}(\beta )^{\ell (\tau (\beta ))}/\mathfrak {S}(\tau (\beta )) \to \mathfrak {M}_{\mathbf {c}}(\alpha )$ whose image equals $\overline {{\mathfrak L}}_{\tau }$ . The resulting map

$$\begin{align*}\prod_{\beta \in \Sigma_{\mathbf{c}}} \mathfrak{M}_{\mathbf{c}}(\beta)^{\ell(\tau(\beta))}/\mathfrak{S}(\tau(\beta)) \to \overline{{\mathfrak L}}_{\tau} \end{align*}$$

is the normalization of $\overline {{\mathfrak L}}_{\tau }$ .

We begin by noting that it follows from the main result of [Reference Crawley-Boevey17] that $\prod _{\beta \in \Sigma _{\mathbf {c}}} \mathfrak {M}_{\mathbf {c}}(\beta )^{\ell (\beta )}$ is normal. Hence the product $\prod _{\beta \in \Sigma _{\mathbf {c}}} \mathfrak {M}_{\mathbf {c}}(\beta )^{\ell (\beta )}/\mathfrak {S}(\tau (\beta ))$ is normal. We will apply the following result.

We construct the morphism

$$ \begin{align*}\prod_{\beta \in \Sigma_{\mathbf{c}}} \mathfrak{M}_{\mathbf{c}}(\beta)^{\ell(\tau(\beta))}/\mathfrak{S}(\tau(\beta)) \to \mathfrak{M}_{\mathbf{c}}(\alpha). \end{align*} $$

First, there is a closed embedding $\prod _{\beta \in \Sigma _{\mathbf {c}}} \mathrm {Rep}(\overline {{\mathsf {Q}}},\beta )^{\ell (\beta )} \to \mathrm {Rep}(\overline {{\mathsf {Q}}},\alpha )$ given by taking direct sum of representations:

$$ \begin{align*}((M_{\beta,i})_{i = 1}^{\ell(\tau(\beta))})_{\beta} \mapsto \bigoplus_{\beta} \bigoplus_{i = 1}^{\ell(\tau(\beta))} M_{\beta,i}^{\oplus \tau(\beta)_i}. \end{align*} $$

If $G(\tau ) = \prod _{\beta \in \Sigma _{\mathbf {c}}} G(\beta )^{\times \ell (\tau (\beta ))}$ is the group acting on the product $\prod _{\beta \in \Sigma _{\mathbf {c}}} \mathrm {Rep}(\overline {{\mathsf {Q}}},\beta )^{\ell (\beta )}$ then $G(\tau ) \subset G(\alpha )$ and the closed embedding is equivariant for $G(\tau )$ . Differentiating this inclusion gives an embedding of Lie algebras $\bigoplus _{\beta \in \Sigma _{\mathbf {c}}} \mathfrak {g}(\beta )^{\oplus \ell (\tau (\beta ))} \hookrightarrow \mathfrak {g}(\alpha )$ . This fits into a commutative diagram

implying that there is a closed embedding of schemes $(\prod \mu _{\beta }^{\ell (\beta )})^{-1}(\mathbf {c}) \hookrightarrow \mu ^{-1}(\mathbf {c})$ . Since this is $G(\tau )$ -equivariant it induces a map

$$ \begin{align*}(\prod \mu_{\beta}^{\ell(\beta)})^{-1}(\mathbf{c}) /\!\!/\! \, G(\tau) \to \mu^{-1}(\mathbf{c}) /\!\!/\! \, G(\alpha). \end{align*} $$

Taking the reduced scheme structure on both sides we get a map $\prod _{\beta \in \Sigma _{\mathbf {c}}} \mathfrak {M}_{\mathbf {c}}(\beta )^{\ell (\beta )} \to \mathfrak {M}_{\mathbf {c}}(\alpha )$ . The group $\prod _{\beta \in \Sigma _{\mathbf {c}}} \mathfrak {S}(\tau (\beta ))$ acts naturally on the left hand side. It is clear on the level of points that the above map factors through the action of this group. However, to see this algebraically, we note that $\prod _{\beta \in \Sigma _{\mathbf {c}}} \mathfrak {S}(\tau (\beta )) \hookrightarrow N_{G(\alpha )}(G(\tau ))/G(\tau )$ and so we get an induced map

$$ \begin{align*}\left( (\prod \mu_{\beta}^{\ell(\beta)})^{-1}(\mathbf{c}) /\!\!/\! \, G(\tau) \right) / \prod_{\beta \in \Sigma_{\mathbf{c}}} \mathfrak{S}(\tau(\beta)) \to \mu^{-1}(\mathbf{c}) /\!\!/\! \, G(\alpha). \end{align*} $$

In other words, this is a morphism $\prod _{\beta \in \Sigma _{\mathbf {c}}} \mathfrak {M}_{\mathbf {c}}(\beta )^{\ell (\beta )}/\mathfrak {S}(\tau (\beta )) \to \mathfrak {M}_{\mathbf {c}}(\alpha )$ . As noted previously, the left hand side is normal. Inside $\mathfrak {M}_{\mathbf {c}}(\beta )^{\ell (\beta )}$ we write $\mathfrak {M}_{\mathbf {c}}(\beta )^{\ell (\beta ),\circ }$ for the (dense) open set consisting of pairwise distinct, and simple, representations. The set of points in $\mathfrak {M}_{\mathbf {c}}(\beta )$ parameterizing simple representations is the open stratum and hence is even dimensional with complement of codimension at least two. This implies that the complement to $\mathfrak {M}_{\mathbf {c}}(\beta )^{\ell (\beta ),\circ }$ in $\mathfrak {M}_{\mathbf {c}}(\beta )^{\ell (\beta )}$ has codimension at least two as well. Notice that $\mathfrak {S}(\tau (\beta ))$ acts freely on this open set. The image of $\prod _{\beta \in \Sigma _{\mathbf {c}}} \mathfrak {M}_{\mathbf {c}}(\beta )^{\ell (\beta ),\circ }$ equals the symplectic leaf ${\mathfrak L}_{\tau }$ . Moreover,

$$\begin{align*}\prod_{\beta \in \Sigma_{\mathbf{c}}} \mathfrak{M}_{\mathbf{c}}(\beta)^{\ell(\beta),\circ} /\mathfrak{S}(\tau(\beta)) \to {\mathfrak L}_{\tau} \end{align*}$$

is a bijection on closed points. Since both domain and image are smooth, this is an isomorphism. Taking affine closures, it induces a morphism

$$ \begin{align*}\prod_{\beta \in \Sigma_{\mathbf{c}}} \mathfrak{M}_{\mathbf{c}}(\beta)^{\ell(\beta)}/\mathfrak{S}(\tau(\beta))\to \overline{{\mathfrak L}}_{\tau}. \end{align*} $$

This must necessarily agree with the previous map.

Then the main result follows from the following.

Example 2.7. Consider the case where $\mathsf {G}$ is the graph with one vertex and one loop. Then $\mathfrak {M}_0(n) \cong \mathbb {C}^{2n} / \mathfrak {S}_n$ and $\Sigma _0 = \{ e_1 \}$ . This means that the strata are labeled by $\tau \colon \{ e_1 \} \to \mathcal {P}$ such that $|\tau (e_1)| = n$ . This is precisely the set of partitions of n. Therefore we think of $\tau = (1^{\tau _1}, 2^{\tau _2},\dots )$ as a partition of n. Then $\mathfrak {M}_0(e_1) = \mathbb {C}^2$ so for a given partition, the normalization of $\overline {{\mathfrak L}}_{\tau }$ is the map

$$ \begin{align*} \mathbb{C}^{2 \ell(\tau)} / \prod_{i \ge 1} \mathfrak{S}_{\tau_i} \to \overline{{\mathfrak L}}_{\tau}. \end{align*} $$

Notice, in particular, that when each part occurs in $\tau $ at most once (this means $\tau _i \le 1$ for all i) then the normalization of $\overline {{\mathfrak L}}_{\tau }$ is just $\mathbb {C}^{2 \ell (\tau )}$ . The leaf closure $\overline {{\mathfrak L}}_{\tau }$ is normal if and only if $\tau = (m^k)$ has a rectangular Young diagram. This can be deduced from [Reference Richardson41, Lemma 2.2], just as in [Reference Richardson41, Proposition 8.1.2] which considers instead the closure of strata in $\mathbb {C}^n / \mathfrak {S}_n$ .

2.9 Normality of a leaf closure

Having identified the normalization of a leaf closure, it is natural to ask if the leaf closure itself is normal. This is addressed in detail in [Reference Bellamy and Schedler5]. In the case of Calogero–Moser varieties, we will see below that the answer is no in general. However, we give one situation (which often occurs in the context of Calogero–Moser varieties) where the leaf closure is normal.

Proof. Note that since $\alpha ^{(i)}$ is real, $\mathfrak {M}_{\mathbf {c}}(\alpha ^{(i)})$ is a single point. Therefore, Theorem 1.9 says that $\iota \colon \mathfrak {M}_{\mathbf {c}}(\beta ) \to \overline {{\mathfrak L}}_{\tau }$ is the normalization map. The morphism $\iota ^* \colon \mathbb {C}[\overline {{\mathfrak L}}_{\tau }] \to \mathbb {C}[\mathfrak {M}_{\mathbf {c}}(\beta )]$ is necessarily injective. We must show that it is surjective. By [Reference Le Bruyn and Procesi26, Theorem 1], both the ring $\mathbb {C}[\mathfrak {M}_{\mathbf {c}}(\beta )]$ and the ring $\mathbb {C}[\mathfrak {M}_{\mathbf {c}}(\alpha )]$ are generated by traces of oriented cycles $\omega $ in $\mathsf {G}$ . If $\omega $ is an oriented cycle starting and ending at vertex j and $X \in \mu ^{-1}(\mathbf {c})$ then we write $\operatorname {{\mathrm {Tr}}} \omega (X)$ for the trace of the endomorphism $\omega (X)$ of $V_j$ ( $\operatorname {{\mathrm {Tr}}} \omega (X)$ does not depend on the choice of j). Let $M_i$ denote the simple $\Pi ^{\mathbf {c}}$ -module of dimension $\alpha ^{(i)}$ and choose some $X_i \in \mu _{\alpha ^{(i)}}^{-1}(\mathbf {c})$ above $M_i$ . Then, as described above, we have a commutative diagram

where $\iota _0(X) = (X \otimes \operatorname {{\mathrm {Id}}}_{n_1} + X_2 \otimes \operatorname {{\mathrm {Id}}}_{n_2} + \cdots + X_k \otimes \operatorname {{\mathrm {Id}}}_{n_k})$ . If $[M] \in \mathfrak {M}_{\mathbf {c}}(\beta )$ and $X \in \mu ^{-1}_{\beta }(\mathbf {c})$ is any (not necessarily semisimple) lift of $[M]$ , then

$$ \begin{align*} (\iota^* \operatorname{{\mathrm{Tr}}} \omega)([M]) & = (\iota_0^* \operatorname{{\mathrm{Tr}}} \omega)(X) \\ & = \operatorname{{\mathrm{Tr}}} \omega(X \otimes \operatorname{{\mathrm{Id}}}_{n_1} +X_2 \otimes \operatorname{{\mathrm{Id}}}_{n_2} + \cdots + X_k \otimes \operatorname{{\mathrm{Id}}}_{n_k}) \\ & = n_1 \operatorname{{\mathrm{Tr}}} \omega(X) + n_2 \operatorname{{\mathrm{Tr}}} \omega(X_2) + \cdots + n_k \operatorname{{\mathrm{Tr}}} \omega(X_k) \\ & = n_1 (\operatorname{{\mathrm{Tr}}} \omega)([M]) + n_2 \operatorname{{\mathrm{Tr}}} \omega(X_2) + \cdots + n_k \operatorname{{\mathrm{Tr}}} \omega(X_k). \end{align*} $$

Hence, $\operatorname {{\mathrm {Tr}}} \omega = \frac {1}{n_1} (\iota ^* \operatorname {{\mathrm {Tr}}} \omega - (n_2 \operatorname {{\mathrm {Tr}}} \omega (X_2) + \cdots + n_k \operatorname {{\mathrm {Tr}}} \omega (X_k)))$ in $\mathbb {C}[\mathfrak {M}_{\mathbf {c}}(\beta )]$ and the result follows.

In terms of framed quiver varieties $\mathfrak {M}_{\mathbf {c}}(\mathbf {w},\alpha )$ , Proposition 2.8 says that if

$$\begin{align*}\tau = (\beta^{(0)},(n_1, \beta^{(1)}; \dots; n_k, \beta^{(k)})) \end{align*}$$

is a decomposition where every $\beta ^{(i)} \in \Sigma _{\mathbf {c}}$ , for $i = 1,\dots , k$ , is a real root then $\overline {{\mathfrak L}}_{\tau } \cong \mathfrak {M}_{\mathbf {c}}(\mathbf {w},\beta ^{(0)})$ is normal.

3.1 The McKay correspondence

Let $\Gamma \subset \mathrm {SL}(2,\mathbb {C})$ be a finite group and $\mathsf {G}$ the simply laced affine Dynkin graph associated to $\Gamma $ via the McKay correspondence. Thus, I is identified with the set of isomorphism classes of irreducible $\Gamma $ -modules in such a way that $0 \in I$ corresponds to the trivial representation. Then $\Delta = \{ e_0, e_1, \dots , e_r \}$ is a set of simple roots for the affine root system $R \subset Q$ . We let $\delta = \sum _{i \in I} \delta _i e_i$ denote the positive root that spans the radical of $( - , -)$ on Q. It satisfies $\delta _0 = 1$ . For the remainder of the article, $W^{\mathrm {aff}} := W_{\mathsf {G}}$ is the affine Weyl group associated to the affine Dynkin diagram defined by $\Gamma $ . Let $R_{\mathbf {c}}$ denote the set of roots $\alpha $ in R such that $\mathbf {c}(\alpha ) = 0$ . We will fix a lift $P \subseteq \mathfrak {h}^*$ satisfying the conditions of the last paragraph of Remark 2.2, which is unique up to automorphism of $\mathfrak {g}$ .

If $\Gamma $ is nontrivial, then roots $e_1, \dots , e_r$ are a set of simple roots for a finite root system $\Phi \subset R$ . This root system is irreducible and $\theta \in \Phi ^+$ denotes the longest positive root. Then $\delta = e_0 + \theta $ . We denote by $\boldsymbol {\Lambda } \subset Q$ the root lattice for $\Phi $ .

If $\Gamma $ is trivial then we define $\delta := e_0$ and $\boldsymbol {\Lambda }=0$ .

Let $L(\Lambda _0)$ be the integrable highest weight module with highest weight $\Lambda _0$ (the basic representation) for the affine Lie algebra $\mathfrak {g}$ with root system R (when $\Gamma $ is trivial, we take the infinite-dimensional Heisenberg Lie algebra as in [Reference Nakajima39]). We recall the properties of $L(\Lambda _0)$ . First, recall that a weight of $L(\Lambda _0)$ is an element $v \in \mathfrak {h}^*$ of the dual of the Cartan subalgebra such that $L(\Lambda _0)_v \neq 0$ , when $\Gamma $ is nontrivial. For $\Gamma $ trivial, we replace $\mathfrak {h}^*$ with the two-dimensional vector space spanned by $\delta =e_0$ and $\Lambda _0$ . Note that $P,Q \subset \mathfrak {h}^*$ are free abelian groups of rank $r+1$ , with sum $P+Q$ a full lattice in $\mathfrak {h}^*$ , which has dimension $r+2$ . A weight v of $L(\Lambda _0)$ is called maximal if $v + \delta $ is not a weight. Finally, for $v = \sum _i v_i \Lambda _i \in P$ , then for every $\lambda \in \mathbb {C}$ , we say that $v+\lambda \delta $ is dominant if $v_i \geq 0$ for all i (otherwise $v+\lambda \delta $ is not dominant).

3.2 Calogero–Moser varieties as quiver varieties

In the case of $\Gamma _n$ with $n> 1$ , there are two types of conjugacy classes of symplectic reflections. Namely, the set

$$\begin{align*}\mathcal{S}_0 = \{ s_{i,j} \gamma_i \gamma_j^{-1} \, | \, 1 \le i \neq j \le n \text{ and } \gamma \in \Gamma \} \end{align*}$$

forms a single conjugacy class and each $\{ \gamma _i \, | \, \gamma \in \mathcal {C}, 1 \le i \le n \}$ is another conjugacy class, as $\mathcal {C}$ runs over all nontrivial conjugacy classes in $\Gamma $ . Therefore, $c = (c_1, \underline {c})$ . Here $\underline {c} \colon \Gamma \smallsetminus \{ 1 \} \to \mathbb {C}$ is a conjugate invariant function and $c_1$ is the value of c on $\mathcal {S}_0$ . From $\underline {c}$ we get the element $z(\underline {c}) \in Z (\mathbb {C} \Gamma )$ by setting $z(\underline {c}) := \sum _{\gamma \in \Gamma \setminus \{ 1 \} } \underline {c}(\gamma )\gamma $ . When $n = 1$ , the conjugacy classes of symplectic reflections are just the $\mathcal {C}$ and $c = (\underline {c})$ ; there is no $c_1$ .

From c, we define a parameter $\mathbf {c} \in \operatorname {{\mathrm {Hom}}}_{\mathbb {Z}}(Q,\mathbb {C})$ by

(3.1) $$ \begin{align} \mathbf{c} (e_i) := -\frac{1}{2} c_1 |\Gamma| \operatorname{{\mathrm{Tr}}}_{S_i} \mathbf{e}_{\Gamma} + \operatorname{{\mathrm{Tr}}}_{S_i} z(\underline{c}), \quad \forall \, 0 \le i \le r. \end{align} $$

Here $S_i$ is the irreducible representation (of dimension $\delta _i$ ) of $\Gamma $ corresponding to vertex i and $\mathbf {e}_{\Gamma }$ is the trivial idempotent in $\mathbb {C} \Gamma $ . When $n = 1$ , one should take an arbitrary $c_1$ in (3.1). The variety ${\mathcal {X}}_{\mathbf {c}}(n\delta ) = \mathfrak {M}_{{\mathbf {c}}}(\Lambda _0, n \delta )$ does not depend on $c_1$ when $n=1$ . The following is [Reference Martino33, Theorem 1.4], generalizing [Reference Etingof and Ginzburg21, Theorem 1.13].

Note that, with our conventions, the theorem still makes sense when $n = 0$ since ${\mathcal {X}}_{{\mathbf {c}}}(0)$ is a point.

The number $\mathbf {c} (\delta )$ is usually called the level. Since we are at $t = 0$ , there are isomorphisms ${\boldsymbol {\mathcal {Z}}}_{c}(\Gamma _n) {\;\stackrel {\sim }{\to }\;} {\boldsymbol {\mathcal {Z}}}_{q c}(\Gamma _n)$ and ${\mathcal {X}}_{{\mathbf {c}}}(n \delta ) {\;\stackrel {\sim }{\to }\;} {\mathcal {X}}_{q{\mathbf {c}}}(n \delta )$ for any $q \in \mathbb {C}^{\times }$ , compatible with the identification of Theorem 3.3. Therefore, we only need to consider the cases $\mathbf {c}(\delta ) = 1$ and $\mathbf {c}(\delta ) = 0$ .

3.3 Parabolic subgroups

A subgroup $\mathrm {P} \subset \Gamma _n$ is said to be a parabolic subgroup if it is the stabilizer of some vector $v \in V$ . Parabolic subgroups play an important role in the classification of symplectic leaves in the Calogero–Moser variety. First, we explicitly describe the parabolic subgroups of $\Gamma _n$ .

Lemma 3.4. The parabolic subgroups of $\Gamma _n$ are, up to conjugacy, of the form $\Gamma _m \times \mathfrak {S}_{\lambda }$ , where $\lambda $ is a partition with $n = |\lambda | + m$ . The normalizer of this parabolic can be described as

$$ \begin{align*} N_{\Gamma_n}(\Gamma_m \times \mathfrak{S}_{\lambda}) = \Gamma_m \times ((\mathfrak{S}_{\lambda} \times \Gamma^{\ell(\lambda)}) \rtimes\, \mathfrak{S}(\lambda)) = \Gamma_m \times N_{\Gamma_{n-m}}(\mathfrak{S}_{\lambda}), \end{align*} $$

and hence $N_{\Gamma _n}(\Gamma _m \times \mathfrak {S}_{\lambda }) / (\Gamma _m \times \mathfrak {S}_{\lambda }) \cong \prod _{i \ge 1} \Gamma _{n_i}$ , where $n_i = | \{ k \, | \, \lambda _k = i \} |$ as in Definition 2.1.

Proof. The fact that the parabolic subgroups are all conjugate to a subgroup of the form $\Gamma _m \times \mathfrak {S}_{\lambda }$ is standard; see, for example, the proof of [Reference Bellamy, Schmitt and Thiel6, Proposition 3.4]. If we let $U \subset (\mathbb {C}^2)^n$ denote the set of points of the form $(0,\dots ,0,v_1,\dots , v_1,v_2,\dots ,v_2,\dots , v_{\ell (\lambda )})$ where $v_i \in \mathbb {C}^2$ with $(\Gamma \cdot v_i) \cap (\Gamma \cdot v_j) = \emptyset $ for $i \neq j$ and there are $\lambda _1$ copies of $v_1$ , $\lambda _2$ copies of $v_2$ etc., then $\Gamma _m \times \mathfrak {S}_{\lambda }$ is the stabilizer of any $u \in U$ and $N_{\Gamma _n}(\Gamma _m \times \mathfrak {S}_{\lambda })$ is the set of elements of $\Gamma _n$ mapping U into itself. Then it is straightforward to check that

$$\begin{align*}N_{\Gamma_n}(\Gamma_m \times \mathfrak{S}_{\lambda}) = \Gamma_m \times ((\mathfrak{S}_{\lambda} \times \Gamma^{\ell(\lambda)}) \rtimes\, \mathfrak{S}(\lambda)). \end{align*}$$

The final isomorphism follows from the fact that $\Gamma ^{\ell (\lambda )} \rtimes\, \mathfrak {S}(\lambda ) = \prod _{i \ge 1} \Gamma _{n_i}$ .

Let $\mathrm {P}$ be a parabolic subgroup of $\Gamma _n$ . By definition, it is normal in its normalizer $N(\mathrm {P})$ . Let $\Xi := N(\mathrm {P})/\mathrm {P}$ be the quotient. The conjugacy class of $\mathrm {P}$ in $\Gamma _n$ is denoted $(\mathrm {P})$ . The algebra ${\mathbf H}_{c}(\Gamma _n)$ has a canonical filtration given by placing $\Gamma _n$ in degree zero and V in degree one. Then ${\mathbf Z}_{c}(\Gamma _n)$ inherits a filtration by restriction and $\mathsf {gr}({\mathbf Z}_{c}(\Gamma _n)) \cong \mathbb {C}[V^*]^{\Gamma _n}$ by [Reference Etingof and Ginzburg21, Theorem 3.3]. If $\mathfrak {p} \subset {\mathbf Z}_{c}(\Gamma _n)$ is the prime ideal defining the closure of a symplectic leaf of ${\boldsymbol {\mathcal {Z}}}_{c}(\Gamma _n)$ , then [Reference Martino31, Theorem 2.8] says that $\mathsf {gr}(\mathfrak {p})$ is a prime ideal defining the closure of a symplectic leaf in $V^*/\Gamma _n$ . Since the leaves of $V^*/\Gamma _n$ are in bijection with conjugacy classes of parabolic subgroups of $\Gamma _n$ , the leaves in ${\boldsymbol {\mathcal {Z}}}_{c}(\Gamma _n)$ can also be labeled by conjugacy classes of parabolic subgroups of $\Gamma _n$ . However, the same conjugacy class can label several different leaves.

This labeling of symplectic leaves by conjugacy classes of parabolic subgroups is important because Losev has shown that there is a notion of induction of leaves whose construction depends on this labeling. Let $\mathrm {PSpec}_{(\mathrm {P})} {\boldsymbol {\mathcal {Z}}}_{c}(\Gamma _n)$ denote the set of all leaves in ${\boldsymbol {\mathcal {Z}}}_{c}(\Gamma _n)$ that are labeled by the conjugacy class $(\mathrm {P})$ . Here $\mathrm {PSpec}$ denotes the set of Poisson prime ideals. We fix a representative $\mathrm {P}$ in $(\mathrm {P})$ . There is a unique zero-dimensional leaf $\{ 0 \}$ in $V^*/\Gamma _n$ ; it is labeled by $(\Gamma _n)$ . Next, we consider the algebra ${\mathbf H}_{c|\mathrm {P}}(\mathrm {P},(V^{\mathrm {P}} )^{\perp })$ , the symplectic reflection algebra defined by the subgroup $\mathrm {P}$ , the restriction $c|\mathrm {P}$ , and the subspace $(V^{\mathrm {P}} )^{\perp } \subset V$ , where the orthogonal is with respect to the symplectic form on V. The group $\Xi $ acts on the Calogero–Moser variety ${\boldsymbol {\mathcal {Z}}}_{c|\mathrm {P}}(\mathrm {P},(V^{\mathrm {P}} )^{\perp })$ , permuting the Poisson prime ideals. The set $\mathrm {PSpec}_{(\mathrm {P})} {\boldsymbol {\mathcal {Z}}}_{c|\mathrm {P}}(\mathrm {P},(V^{\mathrm {P}} )^{\perp })$ of zero-dimensional leaves in $\mathrm {PSpec} \, {\boldsymbol {\mathcal {Z}}}_{c|\mathrm {P}}(\mathrm {P},(V^{\mathrm {P}} )^{\perp })$ is stable under this action. By [Reference Losev27, Theorem 1.3.2(4)]:

Theorem 3.5. There exists a bijection

$$\begin{align*}(\mathrm{PSpec}_{(\mathrm{P})}{\boldsymbol{\mathcal{Z}}}_{\mathbf{c}|\mathrm{P}} (\mathrm{P},(V^{\mathrm{P}} )^{\perp})) / {\Xi} \stackrel{1:1}{\longleftrightarrow} \mathrm{PSpec}_{(\mathrm{P})}{\boldsymbol{\mathcal{Z}}}_{c}(\Gamma_n). \end{align*}$$

We note an immediate corollary.

Proof. By Theorem 3.5, the leaf ${\mathfrak L}$ is labeled by some conjugacy class $(\mathrm {P})$ with $\dim {\mathfrak L} = \dim V^{\mathrm {P}}$ . Lemma 3.4 says that $\mathrm {P} =\Gamma _m \times \mathfrak {S}_{\lambda }$ with $n = m + |\lambda |$ . Moreover, this leaf corresponds to a $\Xi $ -orbit of zero-dimensional leaves in

$$\begin{align*}{\boldsymbol{\mathcal{Z}}}_{\mathbf{c}|\mathrm{P}} (\mathrm{P},(V^{\mathrm{P}} )^{\perp}) = {\boldsymbol{\mathcal{Z}}}_{\mathbf{c}|\Gamma_m} (\Gamma_m,\mathbb{C}^{2m}) \times \prod_{i} {\boldsymbol{\mathcal{Z}}}_{c_1} (\mathfrak{S}_{\lambda_i},\mathfrak{h}_i), \end{align*}$$

where $\mathfrak {h}_i$ is the reflection representation for $\mathfrak {S}_{\lambda _i}$ . We note that

$$ \begin{align*} \mathbf{c} (\delta) & = \sum_{i= 0}^r (\dim S_i) \left( -\frac{1}{2} c_1 |\Gamma| \operatorname{{\mathrm{Tr}}}_{S_i} \mathbf{e}_{\Gamma} + \operatorname{{\mathrm{Tr}}}_{S_i} z(\underline{c}) \right) \\ & = -\frac{1}{2} c_1 |\Gamma| + \operatorname{{\mathrm{Tr}}}_{\mathbb{C} \Gamma} z(\underline{c}) = -\frac{1}{2} c_1 |\Gamma| \end{align*} $$

where we have used the fact that $z(\underline {c})$ is a sum over the nontrivial conjugacy classes in $\Gamma $ . Thus, $\mathbf {c} (\delta ) \neq 0$ implies that $c_1 \neq 0$ . In this case, each ${\boldsymbol {\mathcal {Z}}}_{c_1} (\mathfrak {S}_{\lambda _i},\mathfrak {h}_i)$ is smooth by [Reference Etingof and Ginzburg21, Theorem 1.24]. In particular, it has no zero-dimensional leaves unless $\mathfrak {h}_i = \{ 0 \}$ , that is, $\lambda _i = 1$ . Thus, $\lambda = (1^{n-m})$ and $\mathfrak {S}_{\lambda } = \{ 1 \}$ .

4.1 Roots pairing to zero with $\mathbf {c}$

Recall that $R_{\mathbf {c}}^+$ is the set of roots in $R^+$ that dot to zero with $\mathbf {c}$ . We say that $\alpha \in R_{\mathbf {c}}^+$ is minimal if $\alpha $ cannot be written as a sum of two vectors in $R_{\mathbf {c}}^+$ . Let $\Delta (\mathbf {c}) \subset R_{\mathbf {c}}^+$ be the set of minimal vectors.

We say that a root subsystem $S \subset R$ is a parabolic root subsystem if there exists $\Delta _S \subset \Delta $ such that $\Delta _S$ is a set of simple roots for S.

This means that, up to the action of $W^{\mathrm {aff}}$ , the root system $R_{\mathbf {c}}$ is given by deleting a certain number of nodes in the affine Dynkin diagram. Since $R_{\mathbf {c}}$ is a finite root system, the element w in Lemma 4.2 can be chosen so that $w(\Delta (\mathbf {c})) = w(R_{\mathbf {c}}) \cap \Delta $ .

Recall that we have defined in (2.1) a twisted action of $W^{\mathrm {aff}}$ on Q by

$$\begin{align*}w \star \alpha = w(\alpha - \Lambda_0) + \Lambda_0. \end{align*}$$

Here we have taken $\mathbf {w} = \Lambda _0$ and omitted it from the notation. For $\nu \in \boldsymbol {\Lambda }$ and m a non-negative integer, define

$$ \begin{align*}\gamma(m,\nu) := m \delta + (1/2)(\nu,\nu) \delta - \nu. \end{align*} $$

Lemma 4.3. Let $\alpha \in Q^+$ . Then $\Lambda _0 - \alpha $ is a weight of $L(\Lambda _0)$ if and only if there exists $\nu \in \boldsymbol {\Lambda }$ and $m \ge 0$ such that

$$\begin{align*}\alpha = \gamma(m,\nu). \end{align*}$$

Moreover, if $\Lambda _0 - \alpha $ is a weight of $L(\Lambda _0)$ then there exists $w \in W^{\mathrm {aff}}$ such that $w \star \alpha = m \delta $ .

Recall from Sections 2.3 and 2.5 that we have defined the sets $\Sigma _{\mathbf {c}} \subset R^+_{\mathbf {c}}$ and $E_{\mathbf {c}} \subset Q^+$ (for $\mathbf {w} = \Lambda _0$ ).

4.2 Another presentation of the affine Weyl group

Recall that the affine Weyl group has another presentation. We have $W^{\mathrm {aff}}=W\ltimes \boldsymbol {\Lambda }$ , where W is the nonaffine Weyl group. For each $\beta \in \boldsymbol {\Lambda }$ , denote by $t_\beta $ the image of $\beta $ in $W^{\mathrm {aff}}$ . Each element of $W^{\mathrm {aff}}$ can be written in a unique way in the form $w\cdot t_\beta $ , where $w\in W$ and $\beta \in \boldsymbol {\Lambda }$ . We can also extend the notation $t_\alpha $ to $\alpha \in Q$ by setting $t_\alpha :=t_{\pi (\alpha )}$ for each $\alpha \in Q$ , where $\pi $ is the following map

$$ \begin{align*} \pi\colon Q\to Q/{\mathbb{Z}}\delta\cong \boldsymbol{\Lambda}. \end{align*} $$

Each vector $\alpha \in Q$ defines a linear functional $\overline {\alpha } \in \operatorname {{\mathrm {Hom}}}_{\mathbb {Z}}(Q,\mathbb {C})$ by $\overline {\alpha }(v) := (\alpha ,v)$ . Note that the kernel of the map $Q \to \operatorname {{\mathrm {Hom}}}_{\mathbb {Z}}(Q,\mathbb {C})$ is spanned by $\delta $ .

4.3 The symplectic leaves

We define on Q a quadratic function

(4.1) $$ \begin{align} \varrho(\beta) := \beta_0 + (1/2)(\beta,\beta). \end{align} $$

Write $\Sigma \Sigma _{\mathbf {c}} := \mathbb {N} \Sigma _{\mathbf {c}}$ , which by Lemma 4.4 equals $\mathbb {N} \Delta (\mathbf {c})$ . If $s = |\Delta (\mathbf {c})|$ then Lemma 4.2 implies that $\Sigma \Sigma _{\mathbf {c}} \cong \mathbb {N}^s$ . Define a partial ordering $\succ $ on $\Sigma \Sigma _{\mathbf {c}}$ by $\eta \succ \beta $ if $\eta \neq \beta $ and $\eta - \beta \in \Sigma \Sigma _{\mathbf {c}}$ . Let $\Xi (\mathbf {c})$ denote the set of all vectors $\beta \in \Sigma \Sigma _{\mathbf {c}}$ such that $\eta \in \Sigma \Sigma _{\mathbf {c}}$ and $\eta \succ \beta $ implies that $\varrho (\eta )> \varrho (\beta )$ . In other words, if $\beta \in \Sigma \Sigma _{\mathbf {c}}$ then $\beta $ belongs to $\Xi (\mathbf {c})$ if and only if it is maximal, under the partial ordering $\succ $ , in the set $\{ \eta \in \Sigma \Sigma _{\mathbf {c}} \, | \, \varrho (\eta ) \le \varrho (\beta ) \}$ . Our goal is to prove the following.

Proof. Let ${\mathfrak L}$ be a leaf of ${\mathcal {X}}_{\mathbf {c}}(n \delta )$ . Then, as explained in Section 2.7, ${\mathfrak L}$ is labeled by some decomposition type $\tau = (\beta ^{(0)},(n_1,\beta ^{(1)};\dots; n_k, \beta ^{(k)}))$ . By Lemma 4.3, $\beta ^{(0)} = \gamma (m,\nu )$ for some $\nu \in \boldsymbol {\Lambda }$ and $m \ge 0$ . Moreover, since $n \delta \in E_{\mathbf {c}}$ by Lemma 4.4(ii), we must have $0 \le m \le n$ with $m = n$ if and only if $\tau = (n \delta , (\emptyset ))$ . The roots $\beta ^{(i)}$ for $i \ge 1$ belong to $R^+_{\mathbf {c}}$ . In particular, they are real roots since $\mathbf {c}(\delta ) = -1$ ; this forces $\beta ^{(i)} \neq \beta ^{(j)}$ for $i \neq j$ since ${\mathfrak L} \neq \emptyset $ . We may write

$$ \begin{align*} \beta := n_1 \beta^{(1)} + \cdots + n_k \beta^{(k)} \in \Sigma \Sigma_{\mathbf{c}}. \end{align*} $$

Since $\gamma (m,\nu ) + \beta = n \delta $ , we have $m \delta + (1/2)(\nu ,\nu ) \delta - \nu + \beta = n \delta $ , or $m \delta + (1/2)(\nu ,\nu ) \delta - \nu = n \delta - \beta $ . Taking the coefficient of $e_0$ gives $m + (1/2)(\nu ,\nu ) = n - \beta _0$ and $\nu = \beta - \beta _0 \delta $ . Then

$$ \begin{align*} n - m & = \beta_0 + (1/2)(\nu,\nu) \\ & = \beta_0 + (1/2)(\beta - \beta_0 \delta,\beta - \beta_0 \delta) \\ & = \beta_0 + (1/2)(\beta,\beta) = \varrho(\beta). \end{align*} $$

Hence $0 \le \varrho (\beta ) \le n$ and $m = n - \varrho (\beta )$ . This implies that $\beta $ belongs to the set $\{ \beta \in \Sigma \Sigma _{\mathbf {c}} \, | \, \varrho (\beta ) \le n \}$ . Since $p(\beta ^{(i)}) = 0$ for $i \ge 1$ and

$$\begin{align*}p_{\mathbf{w}}(\beta^{(0)}) = p_{\Lambda_0}(\gamma(m,\nu)) = m = n - \varrho(\beta), \end{align*}$$

the formula (2.4) implies that $\dim {\mathfrak L} = 2m = 2 (n - \varrho (\beta ))$ .

Next we show that $\gamma (m,\nu ) \in E_{\mathbf {c}}$ implies that $\beta \in \Xi (\mathbf {c})$ . If $\beta \notin \Xi (\mathbf {c})$ then there exists $\eta \succ \beta $ in $\Sigma \Sigma _{\mathbf {c}}$ with $\varrho (\eta ) \le \varrho (\beta )$ . Let $m' = n - \varrho (\eta ) \ge 0$ and $\nu ' = \eta - \eta _0 \delta $ so that $\gamma (m',\nu ') + \eta = n \delta $ . Then

$$\begin{align*}\gamma(m,\nu) - \gamma(m',\nu') = (n\delta - \beta) - (n \delta - \eta) = \eta - \beta \in\Sigma \Sigma_{\mathbf{c}}. \end{align*}$$

But this implies that

(4.2) $$ \begin{align} \gamma(m,\nu) = \gamma(m',\nu') + \zeta^{(1)} + \cdots + \zeta^{(s)} \end{align} $$

for some $\zeta ^{(i)} \in \Delta (\mathbf {c})$ . Since $p_{\Lambda _0}(\gamma (m',\nu ')) = m' \ge m = p_{\Lambda _0}(\gamma (m,\nu ))$ and $p(\zeta ^{(i)}) = 0$ , this contradicts the fact that $\gamma (m,\nu ) \in E_{\mathbf {c}}$ . Thus, $\beta \in \Xi (\mathbf {c})$ .

Conversely, assume we have chosen $\beta \in \Xi (\mathbf {c})$ with $\varrho (\beta ) \le n$ . If $\nu := \beta - \beta _0 \delta \in \boldsymbol {\Lambda }$ then

$$\begin{align*}n \delta - \beta = (n - \varrho(\beta)) \delta + \frac{1}{2} (\nu,\nu) \delta - \nu = \gamma(m,\nu) \ge 0, \end{align*}$$

where $m = n - \varrho (\beta )$ . Repeating the argument of the previous paragraphs shows that if $\gamma (m,\nu ) \notin E_{\mathbf {c}}$ then there must exist $m',\nu '$ and $\zeta ^{(i)}$ , with $m' \ge m$ , such that decomposition (4.2) holds. But, again, if we write $\eta = n \delta - \gamma (m',\nu ')$ then $\eta - \beta = \sum _i \zeta ^{(i)} \in \Sigma \Sigma _{\mathbf {c}}$ and hence $\eta \succ \beta $ . Then $\varrho (\eta ) = n - m' \le n - m = \varrho (\beta )$ implies that $\beta \notin \Xi (\mathbf {c})$ .

Part (ii). As shown in Corollary 3.6, the leaves are labeled by $(\Gamma _j)$ for some j when $\mathbf {c}(\delta ) = -1$ . The leaves of dimension $2m$ are all labeled by $(\Gamma _m)$ , so (ii) follows from (i).

By Lemma 3.4, the normalizer of $\Gamma _m$ in $\Gamma _n$ is just $\Gamma _m$ itself. Therefore, in this case, Losev’s induction result says that there is a bijection between the zero-dimensional leaves in ${\boldsymbol {\mathcal {Z}}}_{c}(\Gamma _m)$ and the $2(n-m)$ -dimensional leaves in ${\boldsymbol {\mathcal {Z}}}_{c}(\Gamma _n)$ . It is natural to ask what the relation is between Theorem 4.8 and Losev’s induction result, Theorem 3.5. This is clarified by the following corollary. For $\beta \in \Xi (\mathbf {c})$ with $\varrho (\beta ) \le n$ , write ${\mathfrak L}_n(\beta )$ for the corresponding leaf in ${\boldsymbol {\mathcal {Z}}}_c(\Gamma _n)$ .

The codimension two leaves are particularly easy to enumerate. Note that even though the root system $\Phi $ is irreducible, the system $\Phi _{\mathbf {c}}$ is not irreducible in general.

Example 4.11. Take $\Gamma = \mathbb {Z}/2\mathbb {Z}$ so that $\Gamma _n$ is the Weyl group of type B. Then $Q = \mathbb {Z} e_0 + \mathbb {Z} e_1$ and the real positive roots are

$$\begin{align*}\{ m e_0 + (m+1) e_1 \, | \, m \in \mathbb{Z}_{\ge 0} \}\cup \{ m e_0 + (m-1) e_1 \, | \, m \in \mathbb{Z}_{> 0} \}. \end{align*}$$

Note that we assume $\mathbf {c} (\delta ) = -1$ , which becomes $\mathbf {c}_0 + \mathbf {c}_1 = -1$ . Hence,

$$ \begin{align*} \Delta(\mathbf{c}) = \left\{ \begin{array}{ll} \{ m e_0 + (m+1) e_1 \} & \text{if } \mathbf{c} = (m+1,-m), \, m \in \mathbb{Z}_{\ge 0}, \\ \{ m e_0 + (m-1) e_1 \} & \text{if } \mathbf{c} = (1-m,m), \, m \in \mathbb{Z}_{> 0}, \\ \emptyset & \text{otherwise}. \end{array} \right. \end{align*} $$

Therefore, in the interesting cases, $\Sigma \Sigma _{\mathbf {c}} = \mathbb {N} \alpha $ and $\varrho (k\alpha )=k (k+m)$ . Moreover, $\Xi (\mathbf {c}) = \Sigma \Sigma _{\mathbf {c}}$ in this case. Therefore, the leaves of ${\boldsymbol {\mathcal {Z}}}_{c}(\Gamma _n)$ are in bijection with the set $\{ k \ge 0 \, | \, k(k+m) \le n\}$ .

More generally, if $\Gamma $ is again arbitrary and $R_{\mathbf {c}}$ is a product of rank one root systems, equivalently $(\alpha , \beta ) = 0$ for all $\alpha \neq \beta $ in $\Delta (\mathbf {c})$ , then the above argument shows that the leaves of ${\boldsymbol {\mathcal {Z}}}_{c}(\Gamma _n)$ are in bijection with the set $\{ k \in \mathbb {N}^s \, | \, \sum _i k_i(k_i+m_i) \le n\}$ . Here $m_i$ is the coefficient of $e_0$ in the ith root $\alpha $ in $\Delta (\mathbf {c})$ .

Finally, we describe the closure of leaves in ${\boldsymbol {\mathcal {Z}}}_{c}(\Gamma _n)$ . Recall that $\mathsf {G}'$ is the deframed graph corresponding to $\mathsf {G}$ and $\mathbf {w} = \Lambda _0$ .

Proof. We have

$$ \begin{align*} (e_{\infty} + \gamma(m,\nu),e_i) = \left\{ \begin{array}{ll} -(\nu,e_i) & i \ge 1 \\ -(\nu,e_0) - 1 & i = 0 \\ 2 - (m + (1/2)(\nu,\nu)) & i = \infty. \end{array} \right. \end{align*} $$

Therefore, if $\nu = 0$ then $e_{\infty } + \gamma (m,\nu )$ belongs to the fundamental region if and only if $m> 1$ . Thus, we must show that $e_{\infty } + \gamma (m,\nu )$ is not in the fundamental region if $\nu \neq 0$ .

Consider the case $m = 0$ . Let $v = \Lambda _0 - \frac {1}{2}(\nu ,\nu ) \delta + \nu $ be the corresponding weight for the basic representation $L(\Lambda _0)$ . By Lemma 3.2, it is dominant if and only if $\nu = 0$ . But being dominant means that $(v, e_i) \ge 0$ for all $i = 0, \dots , r$ . Since $(v,e_i) = (\nu , e_i)$ for $i \ge 1$ and $(v,e_0) = (\nu ,e_0) + 1$ , we deduce that the equations $-(\nu ,e_i) \le 0$ for $i \ge 1$ and $-(\nu ,e_0) - 1 \le 0$ force $\nu = 0$ . Thus, $\nu \neq 0$ implies that $e_{\infty } + \gamma (m,\nu )$ is not in the fundamental region $F'$ .

Finally, if $e_{\infty } + \gamma (m,\nu ) = e_{\infty }$ or $e_{\infty } + \delta $ then it is clearly not in the fundamental region.

Proof. As in the proof of Lemma 4.12, we work with dimension vectors on the deframed graph $\mathsf {G}'$ (associated to the framing vector $\mathbf {w} = \Lambda _0$ ). Then $s_i (e_{\infty } + \gamma ) = e_{\infty } + (s_i \star \gamma )$ for $i = 0, \dots , r$ . Let $F'$ denote the fundamental region for $\mathsf {G}'$ . We must show that there is a $\mathbf {c}$ -admissible $w \in W^{\mathrm {aff}} \subset W_{\mathsf {G}'}$ such that $w(e_{\infty } + \gamma (m,\nu )) = e_{\infty } + m \delta $ .

Note that $w(e_{\infty } + \gamma (m,\nu ))$ is a positive root for all $w \in W^{\mathrm {aff}}$ that are $\mathbf {c}$ -admissible since the coefficient of $e_{\infty }$ in $w(e_{\infty } + \gamma (m,\nu ))$ is always one. Let $\beta $ be minimal among all such $w(e_{\infty } + \gamma (m,\nu ))$ under the dominance ordering. We claim that $(\beta ,e_i) \le 0$ for all $i \ge 0$ . Indeed, if $\mathbf {c}' = w^*(\mathbf {c})$ and $\mathbf {c}_i' = 0$ then $(\beta ,e_i) \le 0$ by [Reference Crawley-Boevey15, Lemma 7.2] since $\beta \in \Sigma _{\mathbf {c}_{\beta }'}(\mathsf {G}')$ . Therefore, if $(\beta ,e_i)>0$ then necessarily $\mathbf {c}_i' \neq 0$ . But then $s_i$ is $\mathbf {c}'$ -admissible and $s_i(\beta ) < \beta $ , which contradicts the minimality of $\beta $ . If we write $\beta = e_{\infty } + m' \delta + \frac {1}{2}(\nu ',\nu ') \delta - \nu '$ then $m' = p(\beta ) = p(e_{\infty } + \gamma (m,\nu )) = m$ . This implies that

(4.3) $$ \begin{align} (\beta,e_{\infty}) = 2 - (m + (1/2)(\nu',\nu')). \end{align} $$

If $m> 1$ then we deduce that $\beta $ is in the fundamental region since $(\nu ',\nu ') \ge 0$ . In particular, $\beta \in F' \cap (W_{\mathsf {G}'} \cdot (e_{\infty } + \gamma (m,\nu )))$ . By Lemma 3.2 and Lemma 4.12, $e_{\infty } + m \delta \in F' \cap (W_{\mathsf {G}'} \cdot (e_{\infty } + \gamma (m,\nu )))$ too. But $|F' \cap (W_{\mathsf {G}'} \cdot (e_{\infty } + \gamma (m,\nu )))| \le 1$ by [Reference Kac24, Proposition 3.12(b)]. The claim follows.

If $m = 1$ then equation (4.3) becomes $(\beta ,e_{\infty }) = 1 - (1/2)(\nu ',\nu ')$ . Therefore, $\beta $ is again in $F'$ when $(\nu ',\nu ')> 0$ , that is, when $\nu ' \neq 0$ . But this contradicts Lemma 4.12. If $\nu ' = 0$ then $\beta = e_{\infty } + \delta = e_{\infty } + m \delta $ .

Finally, we consider the case $m = 0$ . Again, if $(\nu ',\nu ')> 2$ then $\beta \in F'$ by (4.3), which contradicts Lemma 4.12. If $(\nu ',\nu ') = 2$ then $\nu '$ is in the finite root system $\Phi $ . But then $(\beta ,e_i) = - (\nu ',e_i) \ge 0$ for $i = 1, \dots , r$ implies that $\nu ' = \theta $ is the highest positive root and hence $\beta = e_{\infty } + e_0$ . But then $(\beta , e_0) = 2-1 = 1$ contradicting the assumption that $(\beta ,e_i) \le 0$ for all $i \ge 0$ . If $\nu ' = 0$ then $\beta = e_{\infty } = e_{\infty } + m \delta $ , as required.

Note that the equation (3.1) defines an isomorphism between the vector spaces of parameters c for the symplectic reflection algebra and the space $\operatorname {{\mathrm {Hom}}}_{\mathbb {Z}}(Q,\mathbb {C})$ . Therefore, it implicitly defines an action of $W^{\mathrm {aff}}$ on the space of parameters, which we also write $c \mapsto w^*(c)$ , for $w \in W^{\mathrm {aff}}$ .

Proof. We let $\nu = \beta - (\Lambda _0,\beta ) \delta $ . Then, by Proposition 4.13, there exists a $\mathbf {c}$ -admissible element $w \in W^{\mathrm {aff}}$ such that $w \star \gamma (m,\nu ) = m \delta $ . Write $\beta = \sum _{i = 1}^s n_i \eta ^{(i)}$ with $\eta ^{(i)} \in \Delta (\mathbf {c})$ . Then the proof of Theorem 4.8 shows that ${\mathfrak L}_n(\beta )$ is the leaf labeled by the representation type

$$\begin{align*}\tau = (\gamma(m,\nu),(n_1,\eta^{(1)};\dots; n_s,\eta^{(s)})). \end{align*}$$

Since every $\eta ^{(i)}$ is a real root, Proposition 2.8 implies that $\overline {{\mathfrak L}}(\beta ) \cong {\mathcal {X}}_{\mathbf {c}}(\gamma (m,\nu ))$ is normal. The fact that w is $\mathbf {c}$ -admissible means that

$$\begin{align*}{\mathcal{X}}_{\mathbf{c}}(\gamma(m,\nu)) \cong {\mathcal{X}}_{w^*(\mathbf{c})}(m \delta) \cong {\boldsymbol{\mathcal{Z}}}_{w^*(c)}(\Gamma_m). \end{align*}$$

Noting that we assume $\mathbf {c}(\delta ) = -1$ , the parameter $w^*(\mathbf {c})$ can be computed explicitly in terms of the vector $\beta $ as follows.

Proof. We continue with the setup of the proof of Theorem 4.14. In particular, $w \in W^{\mathrm {aff}}$ is a $\mathbf {c}$ -admissible element with $w \star \gamma (m,\nu ) = m \delta $ and $\gamma (m,\nu ) + \beta = n \delta $ . Recall from Section 4.2 that w can be written $w = x t_{u}$ , for some (unique) $x \in W$ and $u \in Q$ . Then $w \star \gamma (m,\nu ) = m \delta $ , together with Lemma 4.5, means that there exists $r \in \mathbb {Z}$ such that

$$ \begin{align*} e_{\infty} + m \delta & = w(e_{\infty} + m \delta + (1/2)(\nu,\nu) \delta - \nu) \\ & = x t_u(e_{\infty} + m \delta + (1/2)(\nu,\nu) \delta - \nu) \\ & = x (e_{\infty} + m \delta + (1/2)(\nu,\nu) \delta - \nu - u + r \delta) \\ & = e_{\infty} + (m + r + (1/2)(\nu,\nu)) \delta - x(\nu + u). \end{align*} $$

Comparing the coefficient of $e_0$ in these equalities implies that $m + r + (1/2)(\nu ,\nu ) = m$ , that is, $r = - (1/2)(\nu ,\nu )$ . Then $x(\nu + u) = 0$ , that is, $u = - \nu $ .

Therefore, Lemma 4.6 says that $w^*(\mathbf {c}) = x^* t^*_{-\nu } (\mathbf {c}) = x^*(\mathbf {c} - \mathbf {c}(\delta ) \overline {\nu })$ . Recall that $\nu = \beta - (\Lambda _0,\beta ) \delta $ and hence $\overline {\beta } = \overline {\nu }$ . Since $\mathbf {c}(\delta ) = -1$ , it follows that $\overline {{\mathfrak L}_n(\beta )} \cong {\mathcal {X}}_{x^*(\mathbf {c}')}(m \delta )$ . But ${\mathcal {X}}_{x^*(\mathbf {c}')}(m \delta ) \cong {\mathcal {X}}_{\mathbf {c}'}(m \delta )$ by Remark 4.15.

The partial order on leaves is described as follows.

Proof. For brevity, set $\alpha = e_{\infty } + n \delta $ .

First, we assume that $\eta \succ \beta $ . Then there exist simple $\Pi ^{\mathbf {c}_{n \delta }}$ -modules $M_0,N_0$ and semisimple $\Pi ^{\mathbf {c}_{n \delta }}$ -modules $M_1 = N_{11}$ and $N_{12}$ such that $\dim M_0 = \alpha - \beta , \dim M_1 = \beta , \dim N_{0} = \alpha - \eta $ and $\dim N_{12} = \eta - \beta $ . Let $N_1 = N_{11} \oplus N_{12}$ so that $\dim N_1 = \eta $ . Then $[M_0 \oplus M_1] \in {\mathfrak L}(\beta )$ and $[N_0 \oplus N_{1}] \in {\mathfrak L}(\eta )$ . To show that $\overline {{\mathfrak L}(\beta )} \cap {\mathfrak L}(\eta ) \neq \emptyset $ it suffices, by [Reference Bellamy and Schedler4, Proposition 3.6] and [Reference Bellamy and Schedler4, Corollary 3.25], to argue that the $M_i$ and $N_i$ can be chosen such that $\mathrm {Stab}_G (M_0 \oplus M_1) \subset \mathrm {Stab}_G(N_0 \oplus N_1)$ , where $G = G(\alpha )$ . Since there are no homomorphisms between $M_0$ and the $N_i$ or between $N_0$ and $N_{11},N_{12}$ , we have

$$ \begin{align*} \mathrm{Stab}_G (M_0 \oplus M_1) & = \mathrm{Aut}_{\Pi^{\mathbf{c}}}(M_0) \times \mathrm{Aut}_{\Pi^{\mathbf{c}}}(M_1) \cong \mathbb{C}^{\times} \times \mathrm{Aut}_{\Pi^{\mathbf{c}}}(M_1), \\ \mathrm{Stab}_G (N_0 \oplus N_1) & = \mathrm{Aut}_{\Pi^{\mathbf{c}}}(N_0) \times \mathrm{Aut}_{\Pi^{\mathbf{c}}}(N_{1}) \cong \mathbb{C}^{\times} \times \mathrm{Aut}_{\Pi^{\mathbf{c}}}(M_{1} \oplus N_{12}). \end{align*} $$

The claim follows.

Conversely, if $\overline {{\mathfrak L}(\beta )} \cap {\mathfrak L}(\eta ) \neq \emptyset $ with $\eta \neq \beta $ then there exists $[M] \in {\mathfrak L}(\beta )$ and $[N] \in {\mathfrak L}(\eta )$ such that $\mathrm {Stab}_G(M) \subsetneq \mathrm {Stab}_G(N)$ . We wish to argue that $\eta \succ \beta $ . As in the previous paragraph, we decompose $M = M_0 \oplus M_1$ and $N = N_0 \oplus N_1$ . If $\Delta (\mathbf {c}) = \{ \eta ^{(1)}, \dots , \eta ^{(s)} \}$ and $L(\eta ^{(i)})$ are the simple $\Pi ^{\mathbf {c}}$ -modules with $\dim L(\eta ^{(i)}) = \eta ^{(i)}$ then $M_1 = \bigoplus _{i = 1}^s L(\eta ^{(i)}) \otimes U_i$ , where $\dim U_i = \beta _i$ , and

$$\begin{align*}\mathrm{Stab}_{G(\alpha)}(M) = \mathbb{C}^{\times} \otimes \bigotimes_{i = 1}^s GL(U_i), \end{align*}$$

with first factor $\mathbb {C}^{\times } = \mathrm {Aut}_{\Pi ^{\mathbf {c}}}(M_0)$ . Let $\mathbb {C}^{\times }_{M_1} \subset \mathrm {Stab}_{G(\alpha )}(M)$ be the torus that acts by weight one on $M_1$ and acts trivially on $M_0$ . Let $V_j$ be the vector space at vertex j as in (2.2), with $V_{\infty } = W_0 \cong \mathbb {C}$ since $\alpha _{\infty } = 1$ . Then $\mathrm {Stab}_{G(\alpha )}(M)$ acts on each $V_j$ and $M_1 = \bigoplus _j (M_1 \cap V_j)$ as a $\mathrm {Stab}_{G(\alpha )}(M)$ -module. Notice that

$$\begin{align*}V_j = (M_0 \cap V_j) \oplus (M_1 \cap V_j) = V_j(0) \oplus V_j(1), \end{align*}$$

where $V_j(m)$ is the subspace of $V_j$ of weight m with respect to $\mathbb {C}^{\times }_{M_1}$ . We have $M_0 \cap V_{\infty } = V_{\infty } \cong \mathbb {C}$ because $M_1 \cap V_{\infty } = 0$ . Now, by assumption, $\mathbb {C}^{\times }_{M_1} \subset \mathrm {Stab}_{G(\alpha )}(N)$ . Crucially, $\mathbb {C}^{\times }_{M_1}$ acts trivially on $N_0$ . Indeed, $N_0$ is a simple $\Pi ^{\mathbf {c}}$ -module so $\mathbb {C}^{\times }_{M_1}$ must act as a scalar. But $V_{\infty } = M_0 \cap V_{\infty } = N_0 \cap V_{\infty } \subset N_0$ so $\mathbb {C}^{\times }_{M_1}$ must act trivially on $N_0$ . Therefore, for every $j \ge 0$ , $V_j(0) = M_0 \cap V_j \supset N_0 \cap V_j$ which means that

$$\begin{align*}\alpha_j - \beta_j = \dim M_0 \cap V_j \ge \dim N_0 \cap V_j = \alpha_j - \eta_j \end{align*}$$

and hence $\eta \succeq \beta $ . If $\eta = \beta $ then $\mathrm {Stab}_G(M)$ would be a proper subgroup of $\mathrm {Stab}_G(N)$ which is also conjugate to $\mathrm {Stab}_G(N)$ , which is impossible. Thus, $\eta \succ \beta $ .