Levi-Equivariant Restriction of Spherical Perverse Sheaves

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 $T^*(\check{G}/(\check{U}, \psi_L))$, where the $\textit{Whittaker datum}$ $\psi_L$ is an additive character (determined by $L$) of the maximal unipotent subgroup $\check{U}$.