Elliptic representations for \(\text{Sp}(2n)\) and \(\text{SO}(n)\)
type: Item
instance of: scholarly article
title: Elliptic representations for \(\text{Sp}(2n)\) and \(\text{SO}(n)\)
zbMATH Open document ID: 0797.22007
author: Rebecca A. Herb
published in: Pacific Journal of Mathematics
publication date: 1994-09-26T00:00:00Z
review text: Let \(G\) be a connected, reductive \(p\)-adic group and \(G^ e\) the set of regular elliptic elements. This is an open subset of \(G\). Let \(\pi\) be an irreducible tempered representation of \(G\) with character distribution \(\Theta_ \pi\) and write \(\Theta^ e_ \pi\) for the restriction to \(G^ e\). The representation \(\pi\) is said to be elliptic if \(\Theta^ e_ \pi\) is nonzero. In the line of the philosophy of cusp forms lies a result by \textit{D. Kazhdan} [J. Anal. Math. 47, 1-36 (1986; Zbl 0634.22009)], who showed that an irreducible tempered representation is elliptic just in case it is not an integer linear combination of induced representations. In the paper under consideration this is sharpened for \(G=\text{Sp}(2n)\) and \(G=\text{SO}(2n+1)\) to the assertion that every irreducible tempered representation is either elliptic or is induced from an elliptic representation. It is further shown that this fails for \(G=\text{SO}(2n)\). In this case there are irreducible tempered representations which are not induced representations and are not elliptic.
reviewed by: Anton Deitmar
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 22E35
Mathematics Subject Classification ID: 22D30
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 540674
zbMATH Keywords: reductive \(p\)-adic group
zbMATH Keywords: regular elliptic elements
zbMATH Keywords: irreducible tempered representation
zbMATH Keywords: character distribution
zbMATH Keywords: cusp forms
zbMATH Keywords: induced representations
MaRDI profile type: Publication
full work available at URL: pjm.1993.161.347
OpenAlex ID: W1507117105
DOI: 10.2140/PJM.1993.161.347
description: scientific article; zbMATH DE number 540674