A proof of Harish-Chandra's integrability theorem for cuspidal representations of ${\mathrm{GL}}_n({\mathbb{F}}_{\ell }((t)))$

Abstract

Consider the Chevalley map $$p:{\mathfrak{gl}}_n(F)\to ({\mathfrak{gl}}_n//{\mathrm{GL}}_n)(F),$$ where $F={\mathbb{F}}_{\ell }((t))$. We show that the push forward via $p$ of every smooth compactly supported measure on ${\mathfrak{gl}}_n(F)$ is a measure whose density belongs to $L^q$ for every finite $q$. As a consequence, using the main result of [AGKSc], we obtain local integrability for Harish--Chandra's characters of irreducible cuspidal representations of ${\mathrm{GL}}_n(F)$.

14pp