Sup norm on ${\text{PGL}}_n$ in depth aspect

Abstract

In this short paper we give the sub-local upper bound for the sup norm of an automorphic form on ${\text{PGL}}_n$, whose associated automorphic representation has finite conductor $C(\pi )=p^c$ with $c\to \infty$, and its local component at the place of ramification is a minimal vector belonging to an irreducible representation with generic induction datum.