The endoscopic classification of representations. (English) Zbl 1297.22023
Prasad, D. (ed.) et al., Automorphic representations and \(L\)-functions. Proceedings of the international colloquium, Mumbai, India, January 3–11, 2012. New Delhi: Hindustan Book Agency; Mumbai: Tata Institute of Fundamental Research (ISBN 978-93-80250-49-6/hbk). Studies in Mathematics. Tata Institute of Fundamental Research 22, 1-22 (2013).
From the introduction: The reviewer can do no better than quote almost verbatim from the introduction of this masterly exposition.
(From the introduction:)
We shall outline a classification of automorphic representations of special orthogonal and symplectic groups in terms of those of general linear groups. This necessarily includes a classification of local \(L\)-packets. It also requires a classification of the extended packets that are the local constituents of nontempered automorphic representations. We will restrict to quasisplit orthogonal and symplectic groups \(G\), even though at least some of the results can be extended (not without effort) to inner twists of \(G\).
The methods rest ultimately on two comparisons of trace formulas. One is the spectral identity that is the end product of the stabilization of the trace formula for \(G\). This was established by the author some years ago [Ann. Math. (2) 158, No. 3, 769–873 (2003; Zbl 1051.11027)] under the assumption of the fundamental lemma, which now holds without condition. The other is the spectral identity given by the stabilization of the twisted trace formula for \(GL(N)\). This formula is still conditional. The relevant twisted fundamental lemmas are now known, at least up to the twisted variants of the extensions of the fundamental lemma obtained by Chaudouard and Laumon. The problem is to develop twisted generalizations of the techniques of the above-mentioned paper of the author and of related papers. Until this is done, the results described here have also to be regarded as conditional.
For the entire collection see [Zbl 1293.11002].
(From the introduction:)
We shall outline a classification of automorphic representations of special orthogonal and symplectic groups in terms of those of general linear groups. This necessarily includes a classification of local \(L\)-packets. It also requires a classification of the extended packets that are the local constituents of nontempered automorphic representations. We will restrict to quasisplit orthogonal and symplectic groups \(G\), even though at least some of the results can be extended (not without effort) to inner twists of \(G\).
The methods rest ultimately on two comparisons of trace formulas. One is the spectral identity that is the end product of the stabilization of the trace formula for \(G\). This was established by the author some years ago [Ann. Math. (2) 158, No. 3, 769–873 (2003; Zbl 1051.11027)] under the assumption of the fundamental lemma, which now holds without condition. The other is the spectral identity given by the stabilization of the twisted trace formula for \(GL(N)\). This formula is still conditional. The relevant twisted fundamental lemmas are now known, at least up to the twisted variants of the extensions of the fundamental lemma obtained by Chaudouard and Laumon. The problem is to develop twisted generalizations of the techniques of the above-mentioned paper of the author and of related papers. Until this is done, the results described here have also to be regarded as conditional.
For the entire collection see [Zbl 1293.11002].
Reviewer: Balasubramanian Sury (Bangalore)
MSC:
| 22E55 | Representations of Lie and linear algebraic groups over global fields and adèle rings |
| 11F66 | Langlands \(L\)-functions; one variable Dirichlet series and functional equations |
| 11R37 | Class field theory |
| 20G25 | Linear algebraic groups over local fields and their integers |
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
