Orbital integral bounds the character for cuspidal representations of $GL_n(\mathbb{F}_{\ell}((t)))$

TL;DR

This paper establishes bounds on the character of cuspidal representations of $GL_n(_{ ext{ell}}((t)))$ using orbital integrals, extending Harish-Chandra's integrability theorem to positive characteristic.

Contribution

It proves a local boundedness result for characters in positive characteristic, providing a key step towards a positive characteristic analog of Harish-Chandra's integrability theorem.

Findings

Character of cuspidal representations is locally bounded up to a logarithmic factor.

Orbital integrals provide bounds for characters in positive characteristic.

Foundation for a positive characteristic version of Harish-Chandra's integrability theorem.

Abstract

We prove that the character of an irreducible cuspidal representation of $GL_n(\mathbb{F}_{\ell}((t)))$ is locally bounded up to a logarithmic factor by the orbital integral of a matrix coefficient of this representation. The characteristic $0$ analog of this result is part of the proof of the celebrated Harish-Chandra's integrability theorem. In a sequel work [AGKS] we use this result in order to prove a positive characteristic analog of Harish-Chandra's integrability theorem under some additional assumptions.