Linear independence of intertwining operators.
type: Item
instance of: scholarly article
title: Linear independence of intertwining operators.
zbMATH Open document ID: 1039.22004
author: Dubravka Ban
published in: Journal of Algebra
publication date: 2004-02-03T00:00:00Z
review text: Intertwining operators are important for the trace formula and for the understanding of the reducibility of induced representations. Some of their properties are condensed in the \(R\)-groups. Classically, \(R\)-groups are defined in terms of the Plancherel measure and hence require temperedness. An alternate description is in terms of the \(L\)-group and the Langlands correspondence. Arthur conjectured that in this context, one should be able to define an \(R\)-group, with right basic properties, for certain cases of nontempered representations. Let \(G\) be a split connected reductive \(p\)-adic group, let \(P=MU\) be a standard parabolic subgroup of \(G\), and let \(\sigma\) be a square integrable representation of \(M\). Let \(R\) denote the \(R\)-group corresponding to the parabolically induced representation \(\text{ind}_M^G(\sigma)\). Attached to each element \(r\) in \(R\) is the normalized standard intertwining operator \[ A(\sigma,r)\in\Hom_G(\text{ind}_M^G(\sigma),\text{ind}_M^G(\sigma))=:C(\sigma). \] The set \(\{A(\sigma,r): r\in R\}\) is basis for \(C(\sigma)\). Let \(\widehat\sigma\) be the image of \(\sigma\) by the involution defined in [\textit{A.-M. Aubert}, Trans. Am. Math. Soc. 347, No. 6, 2179--2189 (1995; Zbl 0827.22005) or \textit{P. Schneider} and \textit{U. Stuhler}, Publ. Math., Inst. Hautes Étud. Sci. 85, 97--191 (1997; Zbl 0892.22012)]. Assuming that \(\widehat\sigma\) is unitary, the author has proved in [Ann. Sci. Éc. Norm. Supér., IV. Sér. 35, 673--693 (2002; Zbl 1039.22010)] that \(\text{ind}_M^G(\sigma)\) and \(\text{ind}_M^G(\widehat\sigma)\) have the same intertwining algebras, that is, \(C(\sigma)\simeq C(\widehat\sigma)\) and established a connection between the normalized intertwining operators \(A(\sigma,r)\) and \(A(\widehat\sigma,r)\). In the paper under review, she proves that the set \(\{A(\widehat\sigma,r): r\in R\}\) is basis for \(C(\widehat\sigma)\). In this manner, she defines the \(R\)-group for a class of nontempere…
Mathematics Subject Classification ID: 22E35
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 2037110
zbMATH Keywords: intertwining operator
zbMATH Keywords: Aubert involution
zbMATH Keywords: duality
zbMATH Keywords: \(p\)-adic group
zbMATH Keywords: \(R\) group
reviewed by: Anne-Marie Aubert
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2002.11.004
OpenAlex ID: W2051689880
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DOI: 10.1016/J.JALGEBRA.2002.11.004
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description: scientific article; zbMATH DE number 2037110