On the set of maximal nilpotent supports of supercuspidal representations
type: Item
instance of: scholarly article
title: On the set of maximal nilpotent supports of supercuspidal representations
zbMATH Open document ID: 1321.22024
author: Yujun Qin
published in: Pacific Journal of Mathematics
publication date: 2014-12-30T00:00:00Z
review text: This article makes a valuable contribution to our understanding of the wave-front set of supercuspidal representations of split classical \(p\)-adic groups. According to an important result of \textit{C. Moeglin} and \textit{J. L. Waldspurger} [Math. Z. 196, 427--452 (1987; Zbl 0612.22008)], the wave-front set of an irreducible representation of a \(p\)-adic group, which is a set of nilpotent orbits in the Lie algebra, may be understood either in terms of the Fourier transform of the character distribution, or in terms of degenerate Whittaker functionals. The latter is the view taken in this paper. Thus, the general context is the problem of computing \(\mathcal{N}_{\text{Wh}, \max}(\pi)\), defined as in Moeglin-Waldspurger, for \(\pi\) an irreducible representation of a \(p\)-adic group. The class of representations considered is that of supercuspidals \(\pi(T, \chi)\) attached to pairs \((T, \chi)\) where \(T\) is a minisotropic torus and \(\chi\) is a regular character of \(T\), according to a construction given in [\textit{J. D. Adler}, Pac. J. Math. 185, No. 1, 1--32 (1998; Zbl 0924.22015)]. (The terminology ``minisotropic'' and ``regular'' may be found in [\textit{M. Reeder}, J. Reine Angew. Math. 620, 1--33 (2008; Zbl 1153.22021)]; for semisimple groups, minisotropy reduces to anisotropy.) Such representations were organized into packets in [Reeder, loc. cit.], by varying \((T, \chi)\) in a family corresponding to stable conjugacy of tori. So, the more specific context of the present work is the problem of computing \(\mathcal{N}_{\text{Wh}, \max}(\pi(T, \chi))\), given \(T\) and \(\chi\), and addressing how it varies in an \(L\)-packet. This problem is partly solved in [\textit{S. DeBacker} and \textit{M. Reeder}, Compos. Math. 146, No. 4, 1029--1055 (2010; Zbl 1195.22011)], where it is shown that \(\pi(T, \chi)\) is generic if and only if the torus \(T\) corresponds to a hyperspecial point in the Bruhat-Tits building. The present paper addresses the case wh…
reviewed by: Joseph Hundley
Mathematics Subject Classification ID: 22E50
zbMATH DE Number: 6382635
zbMATH Keywords: supercuspidal representations
zbMATH Keywords: Bruhat-Tits building
zbMATH Keywords: nilpotent orbits
zbMATH Keywords: wave front set
zbMATH Keywords: positive integral depth
MaRDI profile type: Publication
full work available at URL: pjm.2014.269.169
OpenAlex ID: W2064567746
DOI: 10.2140/PJM.2014.269.169
description: scientific article; zbMATH DE number 6382635