On the set of maximal nilpotent supports of supercuspidal representations

Qin Yujun

Vol. 269 (2014), No. 1, 169–198

DOI: 10.2140/pjm.2014.269.169

Abstract

Let $G$ be a quasisplit reductive group over a $p$-adic field $k$, $T$ a maximal unramified anisotropic torus of $G\left(k\right)$, and $\chi$ a character of $T\left(k\right)$ satisfying certain conditions. Assume the residue characteristic $p$ of $k$ is large enough. It was shown by DeBacker and Reeder that the irreducible supercuspidal representation ${\pi }_{\chi }$ of $G\left(k\right)$ associated to $\left(T\left(k\right),\chi \right)$ is generic if and only if $\mathcal{ℬ}\left(T,k\right)$ is a special vertex of $\mathcal{ℬ}\left(G,k\right)$. We compute the set of maximal nilpotent support ${\mathcal{N}}_{wh,max}\left({\pi }_{\chi }\right)$ when $\mathcal{ℬ}\left(T,k\right)$ is not a special point in $\mathcal{ℬ}\left(G,k\right)$.

Keywords

supercuspidal representations, Bruhat–Tits building, nilpotent orbits

Mathematical Subject Classification 2010

Primary: 22E50

Milestones

Received: 25 October 2012

Revised: 23 March 2014

Accepted: 23 March 2014

Published: 15 July 2014