Hecke algebra representations related to spherical varieties

Let $G$ be a connected reductive group over the algebraic closure of a finite field and let $Y$ be a spherical variety for $G$. We consider perverse sheaves on $G$ and on $Y$ which have a weight for the action of a Borel subgroup $B$ and are endowed with an action of Frobenius. This leads to the definition of a ``generalized Hecke algebra'', attached to $G$, and of a module over that algebra, attached to $Y$. 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 $Y$ and $B$-equivariant sheaves, a geometric proof of results which Lusztig and Vogan obtained by representation theoretic means.