A Betti geometric Casselman-Shalika equivalence

TL;DR

This paper establishes a Betti geometric Casselman-Shalika equivalence connecting Iwahori-Whittaker sheaves on the affine Grassmannian with the Satake category, advancing geometric representation theory.

Contribution

It develops a Betti sheaf framework for Iwahori-Whittaker sheaves and proves a new equivalence in geometric representation theory.

Findings

Proved a Betti geometric Casselman-Shalika equivalence.

Developed a theory of Iwahori-Whittaker sheaves in Betti setting.

Connected Iwahori-Whittaker sheaves with the Satake category.

Abstract

Whittaker sheaves are ubiquitous in geometric representation theory; however, their definition requires one to restrict the sheaf-theoretic setting to either \'etale sheaves or $D$-modules. Gaitsgory and Lysenko proposed a solution to this problem called the Kirillov model, which is well-defined for many sheaf theories. In this paper, we advance the study of the Kirillov model in the setting of Betti sheaves with a particular emphasis on developing a theory of Iwahori-Whittaker sheaves on the affine Grassmannian. Using this framework, we prove a Betti geometric Casselman-Shalika equivalence, which relates Iwahori-Whittaker perverse sheaves on the affine Grassmannian with the Satake category.