Twisted endoscopy and the generic packet conjecture
Let \(F\) be a local field and let \(G\) be a connected reductive quasi-split group over \(F\). Conjecturally the set of the isomorphism classes of irreducible tempered representations of \(G(F)\) should be partitioned into a disjoint union of finite subsets, the \(L\)-packets, parametrized by the Langlands parameters for \(G\). In this situation one has the (local) generic packet conjecture: every tempered \(L\)-packet contains a generic element. A second conjecture of great importance concerns the existence and properties of the twisted endoscopic lifting for \(GL(n)\) with respect to an outer automorphism. This should give liftings of tempered \(L\)-packets for many classical groups to irreducible tempered representations of \(GL(n,F)\). In this paper the author supposes expected properties of the twisted endoscopic lifting (in particular, the matching of orbital integrals to stable orbital integrals up to a transfer factor), and then deduces the generic packet conjecture for the classical groups which lift to \(GL(n)\). In order to accomplish this, the author proves a twisted analogue of the results by \textit{F. Rodier} [in: Non-commut. harmon. Anal., Actes Colloq. Marseille-Luminy 1974, Lect. Notes Math. 466, 151-171 (1975; Zbl 0339.22014)] and \textit{C. Mœglin} and \textit{J. L. Waldspurger} [Math. Z. 196, 427-452 (1987; Zbl 0612.22008)] on the dimension of the space of degenerate Whittaker vectors. In an appendix, the author gives a classification of the isomorphism classes of elliptic twisted endoscopic data for \(GL(n)\) over a local or global field of characteristic zero. Because our knowledge of twisted endoscopy is far from complete, the hypotheses allow one to deduce the generic packet conjecture only in the case of \(U(3)\) at the moment. As it happens, a different method based on the relative trace formula gives both the local and global generic packet conjecture for that case [\textit{S. Friedberg, S. Gelbart, H. Jacquet} and \textit{J. Rogawski}, C. R. Acad. Sci., Paris, Sér. I, Math. 329, No. 4, 255-260 (1999; Zbl 0944.22006)]. However, the result here should be applied to a wider class of groups in the future.
scientific article; zbMATH DE number 1760719
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