Rigid inner forms of real and \(p\)-adic groups. (English) Zbl 1393.22009

For a reductive group \(G\) over a local field \(F\), the local Langlands conjecture is a description of the irreducible, admissible representations of \(G(F)\) in terms of the Weil-Deligne group \(W_F'\) and the complex Langlands dual group \(\hat{G}\). Given tempered Langlands parameters \(\phi : W_F' \rightarrow \hat{G}\), the expectation is that the corresponding \(L\)-packets \(\pi_{\phi}^G\) satisfy certain properties like their internal parametrization and endoscopic character identities. Shahidi’s tempered \(L\)-packet conjecture states that for a fixed Whittaker datum \(\vartheta\) of \(G\), the \(L\)-packet \(\pi_{\phi}^G\) should contain a unique generic \(\vartheta\)-representation. There are further conjectural expectations. These are known for connected, quasi-split groups. It is known that the situation is much more complicated for general reductive groups that are not quasi-split.
For this purpose, the author introduces a new type of cohomology set for (affine) algebraic groups by replacing the Galois group with a certain Galois gerbe that is canonically associated to any local field \(F\) of characteristic \(0\). When \(G\) is connected, and reductive over \(F\), and \(Z\) is a finite central subgroup containing the center of the derived subgroup of \(G\), the author defines the cohomology set \(H^1(u \rightarrow W, Z \rightarrow G)\) which is a finite set with some remarkable properties. Here, \(u\) is a certain pro-algebraic group and \(W\) is a certain extension of the absolute Galois group by \(u(\bar{F})\). In particular, \(H^1(\mathrm{Gal},G)\) injects into it and this cohomology set surjects onto \(H^1(\mathrm{Gal},G/Z(G))\). Therefore, every inner twist can be equipped with an element of this cohomology set. The author is able to give a very precise conjectural description of the internal structure of tempered \(L\)-packets and character identities satisfied by them. In particular, he provides a solution to problem 9.3 mentioned by D. A. Vogan jun. [Contemp. Math. 145, 305–379 (1993; Zbl 0802.22005)]. Further, he is able to provide explicit formulae for Arthur’s conjectural spectral transfer factors and mediating functions.
Already, these local results have been applied to global problems. Studying the cohomology of certain Galois gerbes over number fields, the author has provided in later work, a bridge between refined local endoscopy as introduced here, and classical global endoscopy. As particular applications, he has expressed the canonical adelic transfer factor that governs the stabilization of the Arthur-Selberg trace formula as a product of normalized local transfer factors, and gives an explicit constriction of the pairing between an adelic \(L\)-packet and the corresponding \(S\)-group based on the conjectural pairings in the local setting that is the essential ingredient in the description of the discrete automorphic spectrum of a reductive group.
The reader is referred to the author’s nine and a half pages long admirable introduction which gives a very clear description of all subtleties.


22E50 Representations of Lie and linear algebraic groups over local fields


Zbl 0802.22005
Full Text: DOI arXiv


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