On Harish-Chandra's integrability theorem in positive characteristic

TL;DR

This paper investigates Harish-Chandra's integrability theorem for cuspidal representations of $GL_n(F)$ over fields of positive characteristic, establishing integrability under desingularization hypotheses and unconditional regularity when the characteristic exceeds n/2.

Contribution

It extends Harish-Chandra's integrability theorem to positive characteristic fields for certain representations, using desingularization and establishing regularity in specific cases.

Findings

Integrability holds under desingularization assumptions in positive characteristic.

Unconditional regularity established for char(F)>n/2.

Results extend classical theorems to new algebraic settings.

Abstract

The celebrated Harish-Chandra's integrability theorem states that the distributional character of an irreducible smooth representation of a p-adic group $G(F)$ is integrable, that is represented by an $L^1_{loc}(G(F))$ function. Here $F$ is a non-Archimedean local field of characteristic $0$ and $G$ is a reductive algebraic group defined over $F$. In this paper we focus on cuspidal representations of $GL_n(F)$ for a field $F$ of positive characteristic. We show that in this case the integrability holds under the hypothesis of existence of desingularization of (certain) algebraic varieties in positive characteristics. Furthermore, in the case $char(F)>n/2$ we establish the regularity of such characters unconditionally.