On the tempered spectrum of quasi-split classical groups
Let \(F\) denote a \(p\)-adic field of characteristic zero, and let \(\widetilde{G}\) denote the group of \(F\)-points of a symplectic or special orthogonal group defined over \(F\). The authors determine part of the non-discrete tempered spectrum of \(\widetilde{G}\). In order to determine the full spectrum, one would have to understand the reducibility of every representation that is parabolically induced from a discrete series representation \(\sigma\) of a Levi subgroup \(M\) of \(\widetilde{G}\). However, from work of \textit{D. Goldberg} [Am. J. Math. 116, 1101-1151 (1994; Zbl 0851.22021)], one only needs to consider the cases where \(M\) is maximal. Of these, the case where \(M\) is a general linear group has been handled by \textit{F. Shahidi} [Duke Math. J. 66, 1-41 (1992; Zbl 0785.22022)]. Thus, all that remains is the situation where \(M\) has the form \(\text{GL}_n(F)\times G\), where \(G\) is a classical group of the same type as \(\widetilde{G}\). The present paper handles this case when \(\sigma\) is supercuspidal. In order to do so, the authors compute the residue of a standard intertwining operator. Their computation involves a correspondence between conjugacy classes of \(G\) and certain twisted conjugacy classes of \(\text{GL}_n(F)\). Up to a sign, this correspondence agrees with the norm map defined by \textit{R. E. Kottwitz} and \textit{D. Shelstad} [Twisted endoscopy, I: Definitions, norm mappings, and transfer factors, preprint].
scientific article; zbMATH DE number 1425099
scientific article