Commutative character sheaves and geometric types for supercuspidal representations

TL;DR

This paper develops a new framework for geometrizing certain characters of p-adic groups, extending character sheaves to non-commutative cases and applying this to supercuspidal representation types.

Contribution

It introduces a modified category of character sheaves that works for non-commutative groups and applies it to geometrize supercuspidal representation types.

Findings

Established a function-sheaf dictionary for smooth group schemes over finite fields.

Modified character sheaves to address non-commutative groups.

Successfully geometrized supercuspidal representation types.

Abstract

We show that some types for supercuspidal representations of tamely ramified $p$-adic groups that appear in Jiu-Kang Yu's work are geometrizable. To do so, we define a function-sheaf dictionary for one-dimensional characters of arbitrary smooth group schemes over finite fields. In previous work we considered the case of commutative smooth group schemes and found that the standard definition of character sheaves produced a dictionary with a nontrivial kernel. In this paper we give a modification of the category of character sheaves that remedies this defect, and is also extensible to non-commutative groups. We then use these commutative character sheaves to geometrize the linear characters that appear in the types introduced by Jiu-Kang Yu, assuming that the character vanishes on a certain derived subgroup. To define geometric types, we combine commutative character sheaves with Gurevich and Hadani's geometrization of the Weil representation and Lusztig's character sheaves.