Mirabolic affine Grassmannian and character sheaves

TL;DR

This paper computes Frobenius trace functions of mirabolic character sheaves over finite fields, linking them to GL character values and mirabolic Hall-Littlewood basis structure constants.

Contribution

It provides explicit formulas for Frobenius traces of mirabolic character sheaves using symmetric function theory and general linear group characters.

Findings

Frobenius trace functions are expressed via GL character values.

The structure constants relate to the mirabolic Hall-Littlewood basis.

Results connect geometric sheaves with algebraic symmetric functions.

Abstract

We compute the Frobenius trace functions of mirabolic character sheaves defined over a finite field. The answer is given in terms of the character values of general linear groups over the finite field, and the structure constants of multiplication in the mirabolic Hall-Littlewood basis of symmetric functions, introduced by Shoji.