  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 $\check{U}$.
