Levi-Equivariant Restriction of Spherical Perverse Sheaves

Abstract

We study the equivariant cohomology of spherical perverse sheaves on the affine Grassmannian of a connected reductive group $G$ with support in the affine Grassmannian of any Levi subgroup $L$ of $G$. In doing so, we extend the work of Ginzburg and Riche on the $T$-equivariant cofibers of spherical perverse sheaves. We obtain a description of this cohomology in terms of the Langlands dual group $\check{G}$. More precisely, we identify the cohomology of the regular sheaf on ${\mathrm{Gr}}_G$ with support along ${\mathrm{Gr}}_L$ with the algebra of functions on a hyperspherical Hamiltonian $\check{G}$-variety $T^{\ast }(\check{G}/(\check{U},{\psi }_L))$, where the $\text{Whittaker datum}$ ${\psi }_L$ is an additive character (determined by $L$) of the maximal unipotent subgroup $\check{U}$.