Semiorthogonal decompositions of the categories of equivariant coherent sheaves for some reflection groups
Alexander PolishchukUniversity of Oregon, Eugene, USA and National Research University Higher School of Economics, Moscow, Russia
Michel Van den BerghHasselt University, Diepenbeek, Belgium
We consider the derived category of coherent sheaves on a complex vector space equivariant with respect to an action of a finite reflection group . In some cases, including Weyl groups of type , , , , as well as the groups , we construct a semiorthogonal decomposition of this category, indexed by the conjugacy classes of . The pieces of this decompositions are equivalent to the derived categories of coherent sheaves on the quotient-spaces , where is the centralizer subgroup of . 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 , where is a smooth curve.
Cite this article
Alexander Polishchuk, Michel Van den Bergh, Semiorthogonal decompositions of the categories of equivariant coherent sheaves for some reflection groups. J. Eur. Math. Soc. 21 (2019), no. 9, pp. 2653–2749DOI 10.4171/JEMS/890