Keys, C. David L-indistinguishability and R-groups for quasi-split groups: Unitary groups in even dimension. (English) Zbl 0634.22014 Ann. Sci. Éc. Norm. Supér. (4) 20, No. 1, 31-64 (1987). Let G denote the group SU(n,n), with minimal parabolic subgroup \(P=MN\). The author describes the structure of L-packets for the minimal unitary principal series of G; these are the representations ind(\(\lambda\),MN) associated to a Langlands parameter \(\phi\) : \(W_ F\to\) LG. The theory of the R-group, which determines the reducibility of these representations, is recalled, and it is shown that \(R(\lambda)\cong S_{\phi}\) (the group defined in terms of L-group data arising from \(\phi)\). The L-packet \(\Pi_{\phi}\) is defined to be the set of representations of G occurring as components of Ind \(\lambda\), and it is shown there is a pairing (, ): \(S_{\phi}\times \Pi_{\phi}\to {\mathbb{C}}\) so that \[ trace {\mathcal A}(r,\lambda)Ind(\lambda)(f)=\sum_{\pi \in \Pi_{\phi}}<r,\pi >trace\quad \pi (f) \] for \(f\in C_ i^{\infty}(G)\), and A(r,\(\lambda)\) the standard intertwining operator for Ind(\(\lambda)\) corresponding to the element r in \(R(\lambda)\cong S_{\phi}.\) Next the author classifies all possible R-groups for G by constructing a list of characters with non-trivial R-groups, such that any \(\lambda\) with non-trivial R(\(\lambda)\) is conjugate (under the Weyl group) to one on this list. In particular, it is shown that “multiplicity one” fails for G, i.e. examples of non-abelian R(\(\lambda)\) are given. Finally, formulas for the number of components in an L-packet, with multiplicities, are computed. These latter computations involve studying the restrictions of representations of \(\tilde G\) to G, where \(\tilde G=U(n,n)\). Reviewer: S.Gelbart Cited in 3 ReviewsCited in 27 Documents MSC: 22E50 Representations of Lie and linear algebraic groups over local fields 22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods Keywords:parabolic subgroup; L-packets; unitary principal series; Langlands parameter; representations; R-groups; characters × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML References: [1] A. BOREL , Automorphic L-Functions (Proc. Sympos. Pure Math., Vol. 33, part. 2, Amer. Math. Soc., 1979 , pp. 27-61). MR 81m:10056 | Zbl 0412.10017 · Zbl 0412.10017 [2] R. CARTER , Conjugacy Classes in the Weyl Group (Compositio Math., Vol. 25, 1972 , pp. 1-59). Numdam | MR 47 #6884 | Zbl 0254.17005 · Zbl 0254.17005 [3] S. S. GELBART and A. W. KNAPP , L-Indistinguishability and R-Groups for the Special Linear Group . 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