Decomposition of Marsden-Weinstein reductions for representations of quivers (Q2781948)

Let \(Q\) be a quiver. For each arrow of \(Q\) we add an arrow in the opposite direction to obtain the double \(\overline Q\). Let \(\text{Rep}(Q,\alpha)\) be the representation space of \(\alpha\)-dimensional representations. The cotangent bundle of \(\text{Rep}(Q,\alpha)\) can be identified with the representation space \(\text{Rep}(\overline Q,\alpha)\). One can define a group \(G(\alpha)\) such that the \(G(\alpha)\)-orbits in \(\text{Rep}(Q,\alpha)\) correspond to the isomorphy classes of \(\alpha\)-dimensional representations. The group \(G(\alpha)\) acts naturally on \(\text{Rep}(\overline Q,\alpha)\) as well. To this action one can associate a moment map \(\mu_\alpha\colon\text{Rep}(\overline Q,\alpha)\to(\text{Lie }G(\alpha))^*\). If \(\lambda\) is a map from the set of vertices to the base field \(K\), and \(\lambda\) is perpendicular to the dimension vector \(\alpha\), then \(\lambda\) can be identified with an element of \(\text{Lie }G(\alpha)^*\) in a natural way. The \(G(\alpha)\)-orbits in \(\mu_\alpha^{-1}(\lambda)\) correspond to isomorphy classes of \(\alpha\)-dimensional representations of the deformed preprojective algebra \(\Pi^\lambda\). The categorical quotient \(N(\lambda,\alpha):=\mu_\alpha^{-1}(\lambda)/ /G(\alpha)\) is called a Marsden-Weinstein reduction.NEWLINENEWLINENEWLINEIn an earlier paper [Compos. Math. 126, No. 3, 257-293 (2001; Zbl 1037.16007)], the author showed that \(N(\lambda,\alpha)\) is nonempty if and only if \(\alpha\) is a sum of positive roots which are perpendicular to \(\lambda\). Let \(\Sigma_\lambda\) be the set of dimension vectors \(\alpha\) for which the fiber \(\mu_\alpha^{-1}(\lambda)\) contains a simple \(\Pi^\lambda\)-representation. The author gave a combinatorial description of \(\Sigma_\lambda\). He also showed in this earlier paper that \(\mu^{-1}_\alpha(\lambda)\) and \(N(\lambda,\alpha)\) are irreducible if \(\alpha\in\Sigma_\lambda\).NEWLINENEWLINENEWLINEThe main result of this paper shows that if \(\alpha\) does not lie in \(\Sigma_\lambda\), then \(N(\lambda,\alpha)\) can be expressed in terms of Marsden-Weinstein reductions for smaller dimension vectors. This result then shows that \(N(\lambda,\alpha)\) is always irreducible. Another result states that if \(\alpha\in\Sigma_\lambda\) then \(N(\lambda,\alpha)\) is a point if \(\alpha\) is a real root, and isomorphic to a deformation of a Kleinian singularity if \(\alpha\) is isotropic.