Linear independence of intertwining operators.
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 nontempered unitary representations, namely the representations \(\sigma\) such that \(\widehat\sigma\) is square integrable.
scientific article; zbMATH DE number 2037110
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