(Nakajima quiver variety)
Take \(Q\) to be a quiver, \(\alpha\in \mathbb{N}^{Q_0}\) its dimension vector, \(Q_0\) its vertices, \(Q_1\) its arrows, \(\overline{Q}\) the doubled quiver where \(\overline{Q}_1 = Q_1 \sqcup Q_1^*\).
This produces a variety \(\mathfrak{M}_\alpha(Q)\). We can also introduce deformation parameters \(\lambda\in \mathbb{C}^{Q_0}\) and res parameters \(\theta\in \mathbb{Z}^{Q_0}\) or \(\mathbb{Q}^{Q_0}\).)
The set
\[\Sigma_0(Q) = \{\alpha\in R^+ : \alpha = \beta^{(1)} + \dots + \beta^{(k)}, \beta^{(i)}\in R^+ \Rightarrow p(\alpha) > p(\beta^{(1)})+\dots + p(\beta^{(k)}) \} \]
\[\begin{align*} R^+ &= \{\text{roots of } Q\} \\ &= \{\text{Real roots}\} \cup \{\text{Imaginary roots}\} \\ &= \bigcup_i W\cdot e_i \cup W\cdot F_Q \end{align*}\]
where \(W\) is the Weyl group of \(Q\) and \(F_Q\) is the fundamental region, \(\alpha\in \mathbb{N}^{Q_0}\) such that \((\alpha,e_i)\le 0\).
Theorem (Bellamy-S’16) Symplectic leaves of \(\mathfrak{M}_\alpha(Q,\alpha)\) are in bijection with the decompositions \(\alpha = r_1 \beta^{(1)} + \dots + r_k \beta^{(k)}\) with \(\beta^{(i)}\in \Sigma_0(Q)\) and \(r_i\in \mathbb{N}\).
Define
\[\mathcal{L}_{r_1,\beta^{(1)}; \dots; r_k,\beta^{(k)}} = \{\rho\in \mu^{-1}(0) : \rho \cong r_1 \rho^{(1)} + \dots + r_k \rho^{(k)}\}\]
where \(\rho^{(i)}\) is simple non-isomorphic with \(\dim \rho^{(i)}=\beta^{(i)}\). (Note that the non-isomorphic condition implies that real roots can only occur once in the decomposition, which in \(\Sigma_0\) can only be of the form \(e_i\).)
As a corollary, \(\mathfrak{M}_\alpha(Q,\alpha)\) is a union of finitely many locally closed symplectic leaves.
Observe that \(\mathcal{L}_{r,\beta}\subset \overline{\mathcal{L}_{s,\gamma}}\) iff
\[r_1\beta^{(1)} + \dots + r_k \beta^{(k)} \text{ refines } s_1\gamma^{(1)} + \dots + s_l \gamma^{(l)}\]
where \(\beta^{(i)}\in \Sigma_0(Q)\) and \(\gamma^{(j)}\in \Sigma_0(Q)\).
can be deduced from the following result
Theorem (BS draft). We have a classification of \(Q\) and \(\alpha\) where \(\mathfrak{M}_{\alpha}(Q,\alpha)\) has only two leaves.
\(Q\) is \(\tilde{A},\tilde{D},\tilde{E}\)
\(\alpha\) is \(\delta\)
\(Q\) is \(a\xrightarrow{\ge 2} b\)