I might reorganize these information into separate pieces, but now let me put them all together here.
Prop 2.8 in Nakajima’s book on Hilbert Schemes
Prop 6.2 of Le Bruyn
Lemma 2.2 of Punctual Hilbert Schemes for Kleininan Singularities as Quiver Varieties
What we expect: Prop 2.8 + Prop 6.2 should give us the claim that for \(\theta\in F\), every closed orbit in the \(\theta\)-semistable locus of \(\mu^{-1}(0)\) has \(j=0\).
Let \(\Theta_v = \{\theta:\mathbb{Z}^{Q_0}\to \mathbb{Q}: \theta(v)=0\}\) be the linear span of characters, where characters of \(G\), \(\chi_\theta=\prod_i \det(g_i)^{\theta_i}:G\to \mathbb{C}^\times\) are the integral sub-lattice of \(\Theta_v\).
Recall that given parameter \(\theta\in \Theta_v\), a \(\Pi\)-module is \(\theta\)-stable if \(\theta(M)=0\) and \(\theta(M')>0\) for proper subrepresentations. Use \(\ge 0\) for \(\theta\)-semistable.
Two \(\theta\)-semistable modules \(M,N\) are semiequivalent if their orbit closure meet in semistable locus, representation theoretically, this means there are semistable filtrations
\[0=M_0\subset \dots \subset M_{s_1}=M, \quad 0=M_0'\subset \dots\subset M_{s_2}'=M'\]
whose sum of factors are isomorphic \(\bigoplus M_i/M_{i-1} \cong \bigoplus M'_j/M'_{j-1}\).
(This hints us that orbit closures are disjoint unions of orbits, these sum of factors are limits of the biggest orbit.)
Every semiequivalence class has a unique \(\theta\)-polystable representative, which are the direct sum of \(\theta\)-stable modules.
Consider \(\mathcal{M}_\theta = (\mu^{-1}(0)/\!\!/_\theta G)_{red}\), the GIT quotient with reduced scheme structure, it is the categorical quotient of the locus of \(\chi_\theta\)-semistable points of \(\mu^{-1}(0)\) by \(G\). Also identified as the coarse moduli space of \(\theta\)-semistable representations of \(\Pi\) of dimension \(v\).
It is known that every \(\theta\in \Theta_v\) gives an irreducible and normal \(\mathcal{M}_\theta\) with symplectic singularities.
Lemma Let \(\theta\ge \theta'\in \Theta_v\), then \(\pi : \mathcal{M}_\theta \to \mathcal{M}_{\theta'}\) obtained by variation of GIT quotient is a surjective, projective and birational morphism of varieties over \(\mathrm{Sym}^n(\mathbb{C}^2/\Gamma)\).
\(\Theta_v\) are the vectors that have \(0\) paring with \(v\),
\(\Pi\) is the preprojective algebra of the framed McKay quiver, whose ideal is generated by
\[ ij=0, \quad ji + \sum aa^* - \sum a^*a = 0 \]
\(A\) is the algebra \(\Pi\) quotient by \(j\), i.e. the subscheme where \(j=0\).
\(\mathcal{M}_\theta = \mathrm{Rep}(\Pi,v)/\!\!/_\theta G(v)\). It is reduced and irreducible, and is the coarse moduli space of semi-equivalence classes of \(\theta\)-semistable representations of \(\Pi\).
\(F=\{\theta\in \Theta_v : \theta(\delta)>0, \theta(\rho_i)>0, 1\le i\le r\}\), is the fundamental domain.
The full GIT chamber decomposition of \(\Theta_v\) is described in [2], but we can restrict to those characters lie in \(\overline{F}\) by the action of Namikawa-Weyl group. For each chamber \(C\subset F\), variation of GIT quotient defines a projective symplectic (hence crepant) resolution \(\mathcal{M}_C\to \mathcal{M}_0\) of singularities.
Prop. For \(\theta\in F\), The subscheme \(\mathrm{Rep}(A,v)\subset \mathrm{Rep}(\Pi,v)\) gives an isomorphism of schemes \(\mathrm{Rep}(A,v)/\!\!/_\theta G(v) \cong \mathcal{M}_\theta\), as an isomorphism over \(\mathcal{M}_0 = \mathrm{Sym}^n(\mathbb{C}^2/\Gamma)\).
That (seems to) means, if your \(\theta\) is generic and in fundamental domain, any representation in \(\mathcal{M}_\theta\), or \(\theta\)-semistable representation can be represented by a \(\Pi\)-module that has \(j=0\).
Question: Does this mean any \(\theta\)-semistable representation in \(\mathrm{Rep}(\Pi,v)\) is semi-equivalent to a \(\Pi\)-module with \(j=0\)?
For \(\theta\in F\) generic, \(C\) be the GIT chamber containing \(\theta\).
If \(\theta\in C_+\),