Define \(R^+_{\lambda,\theta}=\{\alpha\in R^+ : \lambda\cdot \alpha = \theta\cdot \alpha = 0\}\),
\[ \Sigma_{\lambda,\theta} = \left\{ \alpha \in R^+_{\lambda,\theta} : p(\alpha) > \sum p(\beta^{(i)}) \text{ for all } \alpha = \sum n_i \beta^{(i)} \in R^+_{\lambda,\theta} \right\},\]
these are the dimension vectors that can have representations.
Write
\[\alpha = n_1 \sigma^{(1)} + \dots + n_k \sigma^{(k)}, \]
It is a canonical decomposition if
\(\sigma^{(i)}\)s are distinct
any other decomposition of \(\alpha\) into roots in \(\Sigma_{\lambda,\theta}\) is a refinement of this decomposition.
The canonical decomposition gives isomorphism \[\mathfrak{M}_\lambda(\alpha,\theta) \cong \prod_i \mathrm{Sym}^{n_i}(\mathfrak{M}_\lambda(\sigma^{(i)},\theta))\]
\(\mathfrak{M}_\lambda(\alpha,\theta)\) admits projective symplectic resolution iff each \(\mathfrak{M}_\lambda(\sigma^{(i)},\theta)\) admits projective symplectic resolution.
There exists a \(\theta\)-stable representation of \(\mathrm{Rep}(\Pi^\lambda,\alpha)\) iff \(\alpha\in \Sigma_{\lambda,\theta}\).
The irreducible components of \(\mathrm{Rep}(\Pi_\lambda,v)\) are in bijection with the set of decompositions of \(v\) such that the equality \(p(v) = \sum p(v^{(i)})\) holds, whose dimension will be
\[ 1 + 2 A_Q v\cdot v - v\cdot v .\]
A representation is polystable iff it is a direct sum of stable representations.
A representation \(x\) is canonically \(\theta\)-polystable if \(x = x_1\oplus \dots \oplus x_k\) where each \(x_i\) matches \(\dim x_i = \beta^{(i)}\) is \(\theta\)-stable and \(x_i\neq x_j\) unless \(\beta^{(i)} = \beta^{(j)}\) is a real root, i.e. \(p(\beta^{(i)})=0\).
A point \(x\in \mathfrak{M}_\lambda(\alpha, \theta)\) belongs to the smooth locus iff it is canonically \(\theta\)-polystable.
The quiver variety \(\mathfrak{M}_\lambda(\alpha, \theta)\) is smooth iff
each \(\sigma^{(i)}\) is minimal,
\(\sigma^{(i)}\) isotropic \(\Rightarrow n_i = 1\).
Gwyn Bellamy and Travis Schedler, Symplectic Resolutions of Quiver Varieties.
Victor Ginzburg, Lectures on Nakajima’s quiver varieties.