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

TL;DR

This paper constructs semiorthogonal decompositions of derived categories of equivariant coherent sheaves for certain reflection groups, linking them to quotient spaces and using Springer correspondence and formality results.

Contribution

It introduces new semiorthogonal decompositions for categories of equivariant coherent sheaves associated with specific reflection groups, expanding understanding of their structure.

Findings

Decompositions indexed by conjugacy classes of G.

Equivalence to derived categories on quotient spaces V^g/C(g).

Application of Springer correspondence and formality results.

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.