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

Journal of the European Mathematical Society. 2019. Vol. 21. No. 9. P. 2653–2749.

Polishchuk A., van der Bergh M.

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.

Keywords: semiorthogonal decompositions, полуортогональные разложения

Publication based on the results of: Classical, categorical and noncommutative algebraic geometry, special manifolds and geometry representation theory (2019)