Levi-Equivariant Restriction of Spherical Perverse Sheaves

TL;DR

This paper explores the equivariant cohomology of spherical perverse sheaves on the affine Grassmannian, linking it to the Langlands dual group and hyperspherical Hamiltonian varieties, extending previous work by Ginzburg and Riche.

Contribution

It extends the understanding of equivariant cohomology of spherical perverse sheaves to include Levi subgroup support, connecting it with hyperspherical Hamiltonian varieties of the Langlands dual group.

Findings

Identifies cohomology with functions on a hyperspherical Hamiltonian variety.

Extends Ginzburg and Riche's work to Levi subgroup support.

Provides a geometric description involving the Langlands dual group.

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