Invariant Hilbert schemes and resolutions of quotient singularities

Abstract:

Let G be a classical group (SL_n, GL_n, O_n,...) and X the direct sum of p copies of the standard representation of G and q copies of its dual representation, where p and q are positive integers. We consider the invariant Hilbert scheme, denoted H, which parametrizes the G-stable closed subschemes Z of X such that the coordinate ring k[Z] is isomorphic to the regular representation of G.

         In this talk, we will see that H is a smooth variety when n is small, but that H is generally singular. When H is smooth, the Hilbert-Chow morphism H \rightarrow X//G is a canonical resolution of the singularities of the categorical quotient X//G (=Spec(k[X]^G)). Then it is natural to ask what are the good geometric properties of this resolution (for instance if it is crepant).

         To finish, we will see some analogue results in the symplectic setting, that is, by letting p=q and replacing X by the zero fiber of the moment map. The quotients that we get by doing this are isomorphic to the closures of nilpotent orbits in simple Lie algebras, and the Hilbert-Chow morphism is a resolution of their singularities (sometimes a symplectic one).