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

Raphaël Beuzart-Plessis

Vol. 282 (2016), No. 1, 27–34

DOI: 10.2140/pjm.2016.282.27

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 $G$.

Keywords

$p$-adic groups, supercuspidal representations, cusp forms

Mathematical Subject Classification 2010

Primary: 22E50

Secondary: 11F85

Milestones

Received: 9 June 2015

Revised: 17 August 2015

Accepted: 26 August 2015

Published: 24 February 2016