On descent and the generic packet conjecture
For a quasi-split connected reductive group \( G \) be over a \( p \)-adic field \( F \), the local Langlands conjectures postulate that the set of isomorphism classes of irreducible tempered representations of \( G(F) \) can be partitioned into finite subsets, called tempered \( L \)-packets, in a certain natural way. The generic packet conjecture (also called tempered packet conjecture in the literature), due to \textit{F. Shahidi} [Ann. Math. (2) 132, No. 2, 273--330 (1990; Zbl 0780.22005)], says that every such tempered \( L \)-packet should contain a generic representation. In other words, suppose \(\Pi \) is a tempered \(L\)-packet of representations of \( G(F) \), \( B=TU \) a Levi decomposition of a Borel subgroup \( B \) of \( G \), and \(\lambda \) a character of \( U(F) \) which is nondegenerate, i.e., whose stabilizer in \( T(F) \) is the center of \( G(F) \). Then there should exist at least one representation \((\pi,V) \) in \(\Pi \) which is \( \lambda \)-generic, i.e., has a nonzero space of \( (U(F),\lambda) \)-coinvariants. A stronger form of this conjecture says that (for any choice of \( (U(F), \lambda) \) as above) every tempered \( L \)-packet contains precisely one \(\lambda\)-generic representation. This stronger form is known to be true for the general linear group \( \mathrm{GL}_n \) over \( F \) (and hence also for \( \mathrm{SL}_n \)). The weaker form of this conjecture for quasi-split symplectic, special orthogonal and unitary groups over \( F \) is addressed by \textit{T. Konno} [Isr. J. Math. 129, 253--289 (2002; Zbl 1004.22008)], assuming the existence of endoscopic character identities, recently established by \textit{J. Arthur} [The endoscopic classification of representations. Orthogonal and symplectic groups. Providence, RI: American Mathematical Society (AMS) (2013; Zbl 1310.22014)] and \textit{C. P. Mok} [Endoscopic classification of representations of quasi-split unitary groups. Providence, RI: American Mathematical Society (AMS) (2015; Zbl 1316.22018)]. The strong form of the conjecture too is known in many cases, including those of quasi-split special orthogonal and symplectic groups. But for these and a few other examples, the generic packet conjecture remains largely unknown (indeed the Langlands parametrization itself is largely so, but one has not been able to conclude the generic packet conjecture from expected properties of the correspondence). For instance, it is not known in the case of a general spin (GSpin) group, either in its strong or weak form. The principal purpose of this paper is to strengthen and generalize the result of Konno [loc. cit.]. Precisely, let \( G \) be a quasi-split group over \( F \), \( \theta \) an automorphism of \( G \) of finite order that preserves an \( F \)-splitting of \( G \), and \( \mathbf{a} \) an element of \( H^1(W_F , Z(\widehat{G})) \), \( Z(\widehat{G}) \) denoting the center of the Langlands dual of \( G \). Let \( \omega \) be the character on \( G(F) \) associated to \( \mathbf{a} \). Consider any group \( H \) that forms part of an endoscopic datum associated to the triplet \( (G,\theta, \mathbf{a}) \), and a \( z \)-extension \( H_1 \) of \( H \) as required by the formalism of twisted endoscopy in general. In this setting the author proves that if a tempered \( L \)-packet \( \Sigma \) on \( H_1(F) \) transfers twisted endoscopically to a tempered \( L \)-packet \( \Pi \) on \( G(F) \), and if \( \Pi \) satisfies the strong form of the generic packet conjecture, then so does \( \Sigma \). In doing so, the author makes stronger assumptions on the character identities than the one in [loc. cit.], but this more precise form of the identities is expected to hold, and has been proved for quasi-split symplectic, special orthogonal and unitary groups. This allows the author to deduce the strong version of the generic packet conjecture for a more general class of groups including general spin (GSpin) groups, if the appropriate conjectural character identities are granted. Moreover, the assumption in [loc. cit.] that the residual characteristic of \( F \) is odd is removed in this work. To conclude, this article is well-written, and the review we have given of it is a paraphrasing of some paragraphs from the introductory section. The general approach to the proof is succinctly described in the introduction, and the basic notions to understand the heart of the article are recalled, with a brief review of twisted endoscopy and semisimple descent.
scientific article; zbMATH DE number 7010461
scientific article