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

Consider the Chevalley map $$p:{\mathfrak{g}\mathfrak{l}}_n(F)\to ({\mathfrak{g}\mathfrak{l}}_n\mathrm{/}\mathrm{/}{\mathrm{G}\mathrm{L}}_n)(F),$$ where $F={\mathbb{F}}_{\mathrm{\ell }}((t))$. We show that the push forward via $p$ of every smooth compactly supported measure on ${\mathfrak{g}\mathfrak{l}}_n(F)$ 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 ${\mathrm{G}\mathrm{L}}_n(F)$.