Geometric Casselman-Shalika in mixed characteristic

TL;DR

This paper develops a geometric analog of the Casselman-Shalika formula for split reductive groups over mixed characteristic local fields, constructing sheaves on the Witt vector affine Grassmannian and analyzing their cohomology.

Contribution

It introduces a geometric framework for the Casselman-Shalika formula in mixed characteristic, linking sheaves on the affine Grassmannian to Fourier coefficients of Hecke operators.

Findings

Constructed sheaves on Witt vector affine Grassmannian

Computed the cohomology of these sheaves

Established the geometric Casselman-Shalika formula in mixed characteristic

Abstract

We establish a geometric analog of the Casselman-Shalika formula for a split connected reductive group over a mixed characteristic local field. In particular, we construct sheaves on the Witt vector affine Grassmannian which geometrize the Fourier coefficients of spherical Hecke operators, and compute their cohomology.