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

2015/03/13 by Alexander Polishchuk, Michel Van den Bergh, Polishchuk, Alexander +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1503.04160

openalex publication_date 2015/03/13 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

Abstract

We consider the derived category of coherent sheaves on a complex vector\nspace equivariant with respect to an action of a finite reflection group G. In\nsome cases, including Weyl groups of type A, B, G₂, F₄, as well as the groups\nG(m,1,n), we construct a semiorthogonal decomposition of this category, indexed\nby the conjugacy classes of G. The pieces of this decompositions are equivalent\nto the derived categories of coherent sheaves on the quotient-spaces V^g/C(g),\nwhere C(g) is the centralizer subgroup of g in G. In the case of the Weyl\ngroups the construction uses some key results about the Springer\ncorrespondence, due to Lusztig, along with some formality statement\ngeneralizing a result of Deligne. We also construct global analogs of some of\nthese semiorthogonal decompositions involving derived categories of equivariant\ncoherent sheaves on Cⁿ, where C is a smooth curve.\n

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