Cuspidal representations associated to \((\text{GL}(n),O(n))\) over finite fields and \(p\)-adic fields
type: Item
instance of: scholarly article
title: Cuspidal representations associated to \((\text{GL}(n),O(n))\) over finite fields and \(p\)-adic fields
zbMATH Open document ID: 0918.22013
author: Jeffrey Hakim
author: Zhengyu Mao
published in: Journal of Algebra
publication date: 1999-03-30T00:00:00Z
review text: Let \(G\) be a finite group and \(H\) a proper subgroup. \((G,H)\) is called a Gelfand pair if \(\text{Ind}^G_H1\) is a multiplicity free representation. Gelfand pairs of real or \(p\)-adic groups are similarly defined. If \({\mathbf G}\) is a reductive group defined over a finite field or a local field \(F\) and \({\mathbf H}\) an algebraic subgroup then \({\mathbf h}\) is called spherical [cf. \textit{D. Luna} and \textit{T. Vust}, Comment. Math. Helv. 58, 186--245 (1983; Zbl 0545.14010)] if there exists a Borel subgroup \({\mathbf B}\) such that \({\mathbf B}{\mathbf H}\) is open in \({\mathbf G}\). If \({\mathbf H}\) is a spherical subgroup then the groups of rational points, \(G= {\mathbf G}(F)\) and \(H={\mathbf H}(F)\), are expected to form a Gelfand pair or almost a Gelfand pair. Here ``almost'' is not a precise term, but taken to mean that ``most'' representations occur with multiplicity one or with small multiplicity in \(\text{Ind}^G_H1\). In the paper under review, the authors study the spherical pair \((\text{GL}(n),O(n))\), where \(O(n)\) is an orthogonal group associated to a symmetric form for the case, where \(F\) is a finite field or a \(p\)-adic field. They also study the related spherical pair \((\text{GL}(n),\text{GO}(n))\), where \(\text{GO}(n)\) is the similitude group. In the case, where \(F= {\mathbb R}\) and \(O(n)\) is the standard compact group, then \((\text{GL}(n,{\mathbb R}),O(n,{\mathbb R}))\) is a Gelfand pair [cf. \textit{I. G. Macdonald}, Symmetric functions and Hall polynomials, 2nd ed. Oxford (1995; Zbl 0824.05059)]. However, when \(F\) is a finite field or a \(p\)-adic field this is no longer true. The authors study the multiplicities of cuspidal representations in \(\text{Ind}^G_H1\) for the finite field case and of ``level zero'' supercuspidal representations for the \(p\)-adic field case. Level zero supercuspidals are those supercuspidals which are obtained from cuspidal representations on the finite group via induction from …
reviewed by: Ehud Moshe Baruch
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 20C33
zbMATH DE Number: 1270876
zbMATH Keywords: Gelfand pair
zbMATH Keywords: spherical pair
zbMATH Keywords: finite field
zbMATH Keywords: \(p\)-adic field
zbMATH Keywords: similitude group
zbMATH Keywords: supercuspidal representations
MaRDI profile type: Publication
full work available at URL: jabr.1998.7664
OpenAlex ID: W2067035859
cites work: Q4110092
cites work: Representations of reductive groups over finite fields
cites work: Metaplectic correspondence
cites work: Q3958534
cites work: Supercuspidal representations of \(\text{GL}(n)\) distinguished by a unitary subgroup
cites work: Q3828486
cites work: Multiplicity-free permutation representations of finite linear groups
cites work: Q3986689
cites work: Q3972765
cites work: Q3947818
cites work: A fundamental lemma for metaplectic correspondence
cites work: Q3708857
cites work: Opération de groupes réductifs dans un type de cônes presque homogènes
DOI: 10.1006/JABR.1998.7664
description: scientific article; zbMATH DE number 1270876