Semiorthogonal decompositions of the categories of equivariant coherent sheaves for some reflection groups

Abstract

We consider the derived category of coherent sheaves on a complex vector space equivariant with respect to an action of a finite reflection group G. In some cases, including Weyl groups of type A, B, G_2, F_4, as well as the groups G(m,1,n), we construct a semiorthogonal decomposition of this category, indexed by the conjugacy classes of G. The pieces of this decompositions are equivalent to the derived categories of coherent sheaves on the quotient-spaces V^g/C(g), where C(g) is the centralizer subgroup of g in G. In the case of the Weyl groups the construction uses some key results about the Springer correspondence, due to Lusztig, along with some formality statement generalizing a result of Deligne. We also construct global analogs of some of these semiorthogonal decompositions involving derived categories of equivariant coherent sheaves on C^n, where C is a smooth curve.