Hecke algebra representations related to spherical varieties

Abstract

Let G G be a connected reductive group over the algebraic closure of a finite field and let Y Y be a spherical variety for G G . We consider perverse sheaves on G G and on Y Y which have a weight for the action of a Borel subgroup B B and are endowed with an action of Frobenius. This leads to the definition of a “generalized Hecke algebra”, attached to G G , and of a module over that algebra, attached to Y Y . The same algebra and the same module can also be defined using constructible sheaves. Comparison of the two definitions gives, in the case of a symmetric variety Y Y and B B -equivariant sheaves, a geometric proof of results which Lusztig and Vogan obtained by representation theoretic means.