A method of proving non-unitarity of representations of \(p\)-adic groups. I
type: Item
instance of: scholarly article
title: A method of proving non-unitarity of representations of \(p\)-adic groups. I
zbMATH Open document ID: 1215.22009
author: Marcela Hanzer
author: Marko Tadić
published in: Mathematische Zeitschrift
publication date: 2011-01-03T00:00:00Z
review text: For a connected reductive group \(G\) over a \(p\)-adic field \(F\), the Steinberg representation \(St\) and its Aubert dual, the trivial representation, are both unitary. Casselman proved that all other irreducible representations having the same infinitesimal character as \(St\) are non-unitary. In this paper, the authors give a partial generalization of Casselman's result. They consider generalized Steinberg representations (defined below) of \(G_\ell=Sp(2\ell,F)\) or \(SO(2\ell+1,F)\). Let \(\nu = |\det|_F\). Let \(\rho\) be an irreducible unitary cuspidal representation of \(GL(k,F)\) and \(\sigma\) an irreducible cuspidal representation of \(G_\ell\). Suppose that \(\text{Ind}^{G_{k+\ell}}(\nu^\alpha \rho \otimes \sigma)\) reduces for some \(\alpha \in (1/2)\mathbb{Z}_{>0}\). Then the representation \[ \Pi= \text{Ind}^{G_{(n+1)k+\ell}}(\nu^{\alpha + n} \rho \otimes \nu^{\alpha + n-1} \rho \otimes \cdots \otimes \nu^{\alpha } \rho \otimes \sigma) \] has the unique irreducible subrepresentation \(\pi\) and the unique irreducible quotient \(\pi'\). The representation \(\pi\) is square-integrable (and hence unitary); it is called a generalized Steinberg representation. The representation \(\pi'\) is the Aubert involution of \(\pi\). This paper and its sequel prove that all irreducible subquotients of \(\Pi\) different from \(\pi\) and \(\pi'\) are not unitarizable. The method of proof is based on the fact that the representation induced from a unitary representation is unitary, and therefore semi-simple.
reviewed by: Dubravka Ban
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 22E35
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 5831050
zbMATH Keywords: classical \(p\)-adic groups
zbMATH Keywords: generalized Steinberg representation
zbMATH Keywords: unitary representations
zbMATH Keywords: non-unitarity criterion
MaRDI profile type: Publication
full work available at URL: s00209-009-0542-7
OpenAlex ID: W1982669965
cites work: Dualite Dans Le Groupe De Grothendieck De La Categorie Des Representations Lisses De Longueur Finie D'un Groupe Reductif p-Adique
cites work: Erratum à “Dualité dans le groupe de Grothendieck de la catégorie des représentations lisses de longueur finie d’un groupe réductif p-adique”
cites work: Q4950682
cites work: Degenerate principal series for even-orthogonal groups
cites work: Induced representations of reductive ${\germ p}$-adic groups. I
cites work: Q3669664
cites work: The unitarizability of the Aubert dual of strongly positive square integrable representations
cites work: Asymptotic properties of unitary representations
cites work: On supports of induced representations for symplectic and odd-orthogonal groups
cites work: Construction of discrete series for classical 𝑝-adic groups
cites work: Induced representations and classification for $GSp(2,F)$ and $Sp(2,F)$
cites work: Representation theory and sheaves on the Bruhat-Tits building
cites work: A proof of Langlands' conjecture on Plancherel measures; complementary series for \(p\)-adic groups
cites work: Twisted endoscopy and reducibility of induced representations for \(p\)- adic groups
cites work: Special representations of reductive p-adic groups are not integrable
cites work: Classification of unitary representations in irreducible representations of general linear group (non-Archimedean case)
cites work: Structure arising from induction and Jacquet modules of representations of classical \(p\)-adic groups
cites work: On regular square integrable representations of p -adic groups
cites work: Induced representations of reductive ${\germ p}$-adic groups. II. On irreducible representations of ${\rm GL}(n)$
DOI: 10.1007/S00209-009-0542-7
description: scientific article; zbMATH DE number 5831050