Vol. 282, No. 1, 2016

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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

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.

$p$-adic groups, supercuspidal representations, cusp forms
Mathematical Subject Classification 2010
Primary: 22E50
Secondary: 11F85
Received: 9 June 2015
Revised: 17 August 2015
Accepted: 26 August 2015
Published: 24 February 2016
Raphaël Beuzart-Plessis
Department of Mathematics
National University of Singapore
Block 17, 10 Lower Kent Ridge Road
Singapore 119076