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

We consider the derived category DGb(V)D^b_G(V) of coherent sheaves on a complex vector space VV equivariant with respect to an action of a finite reflection group GG. In some cases, including Weyl groups of type AA, BB, G2G_2, F4F_4, as well as the groups G(m,1,n)=(μm)nSnG(m,1,n)=(\mu_m)^n\rtimes S_n, we construct a semiorthogonal decomposition of this category, indexed by the conjugacy classes of GG. The pieces of this decompositions are equivalent to the derived categories of coherent sheaves on the quotient-spaces Vg/C(g)V^g/C(g), where C(g)C(g) is the centralizer subgroup of gGg\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 CnC^n, where CC 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