A short proof of the existence of supercuspidal representations for all reductive $p$-adic groups

TL;DR

This paper provides a concise, self-contained proof that every reductive p-adic group admits supercuspidal complex representations, offering an alternative to existing methods based on Deligne-Lusztig theory.

Contribution

It introduces a new, simpler proof leveraging Harish-Chandra theory and elliptic maximal tori, differing from prior approaches by Kret.

Findings

Every reductive p-adic group admits supercuspidal representations.

The proof is shorter and self-contained compared to previous methods.

Relies on the existence of elliptic maximal tori in G.

Abstract

Let $G$ be a reductive $p$-adic group. We give a short proof of the fact that $G$ always admits supercuspidal complex representations. This result has already been established by A. Kret using the Deligne-Lusztig theory of representations of finite groups of Lie type. Our argument is of a different nature and is self-contained. It is based on the Harish-Chandra theory of cusp forms and it ultimately relies on the existence of elliptic maximal tori in $G$.