When does the zero fiber of the moment map have rational singularities?

Let G be a complex reductive group and V a G-module. There is a natural moment mapping μ V⊕ V^*^* and we denote μ⁻¹(0) (the shell) by N. We find criteria for N to have rational singularities and for the categorical quotient $N/\!\!/ G$ to have symplectic singularities, the latter results improving upon [HSS20]. It turns out that for most G-modules V, the shell N has rational singularities. For the case of direct sums of classical representations of the classical groups, N has rational singularities and $N/\!\!/ G$ has symplectic singularities if N is a reduced and irreducible complete intersection. Another important special case is V=pg (the direct sum of p copies of the Lie algebra of G) where p≥ 2. We show that N has rational singularities and that $N/\!\!/ G$ has symplectic singularities, improving upon results of [Bud19] and [AA16]. Let π=π₁(Σ) where Σ is a closed Riemann surface of genus p≥ 2. Let G be semisimple and let Hom(π,G) and X\!(π,G) be the corresponding representation variety and character variety. We show that Hom(π,G) is a complete intersection with rational singularities and that X\!(π,G) has symplectic singularities. If $p>2$ or G contains no simple factor of rank $1$, then the singularities of Hom(π,G) and X\!(π,G) are in codimension at least four and Hom(π,G) is locally factorial. If, in addition, G is simply connected, then X\!(π,G) is locally factorial.