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

Abstract

We consider the derived category $D^b_G(V)$ of coherent sheaves on a complex vector space $V$ 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 $G(m,1,n)=(\mu_m)^n\rtimes S_n$, 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 [23]. 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.