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

Alexander Polishchuk; Michel Van den Bergh

doi:10.4171/jems/890

JournalsjemsVol. 21, No. 9pp. 2653–2749

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

Alexander Polishchuk
University of Oregon, Eugene, USA and National Research University Higher School of Economics, Moscow, Russia
Michel Van den Bergh
Hasselt University, Diepenbeek, Belgium

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Abstract

We consider the derived category $D_{G}^{b} (V)$ 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 $G (m, 1, n) = (μ_{m})^{n} ⋊ S_{n}$ , 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.

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. 2653–2749

DOI 10.4171/JEMS/890