Rigid inner forms of real and \(p\)-adic groups
type: Item
instance of: scholarly article
title: Rigid inner forms of real and \(p\)-adic groups
zbMATH Open document ID: 1393.22009
author: Tasho Kaletha
published in: Annals of Mathematics. Second Series
publication date: 2016-12-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/1304.3292
reviewed by: Balasubramanian Sury
Mathematics Subject Classification ID: 22E50
zbMATH DE Number: 6662220
zbMATH Keywords: local Langlands
zbMATH Keywords: endoscopic transfer
zbMATH Keywords: tempered \(L\)-packet
zbMATH Keywords: \(p\)-adic groups
MaRDI profile type: Publication
OpenAlex ID: W2963514825
arXiv ID: 1304.3292
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DOI: 10.4007/ANNALS.2016.184.2.6
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review text: 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.NEWLINENEWLINEFor 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…