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The Geometry and Cohomology of Some Simple Shimura Varieties.

Let G be an inner form of a general linear group over a non-archimedean local field. We prove that the local Langlands correspondence for G preserves depths. We also show that the local Langlands correspondence for inner forms of special linear groups preserves the depths of essentially tame Langlands parameters.

Progress in Mathematics

Here, published for the first time, are the complete proofs of the fundamental arithmetic duality theorems that have come to play an increasingly important role in number theory and arithmetic geometry. The text covers these theorems in Galois cohomology, tale cohomology, and flat cohomology and addresses applications in the above areas. The writing is expository and the book will serve as an invaluable reference text as well as an excellent introduction to the subject.

Arithmetic duality theorems

analysis of p-adic reductive groups. Our first result, Theorem 7.9, proves a conjecture of Langlands on normalization of intertwining operators by means of local Langlands root numbers and L-functions, at least when the group is quasi-split and the inducing representation is generic. Assuming two natural conjectures in harmonic analysis of p-adic groups, we also prove the validity of the conjecture in general (Theorem 9.5). As our second result we obtain all the complementary series and special representations of quasi-split p-adic groups coming from rank-one parabolic subgroups and generic supercuspidal represen

A proof of Langland’s conjecture on Plancherel measures; Complementary series of $p$-adic groups

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/63B506EB85D6875966CE63BA0336BC2F/S0025557200190515a.pdf/div-class-title-span-class-italic-lie-groups-and-lie-algebras-chapters-1-3-span-by-n-bourbaki-pp-450-dm-98-1989-isbn-3-540-50218-1-springer-div.pdf

Lie groups and Lie algebras (chapters 1-3 ), by N. Bourbaki. Pp 450. DM 98. 1989. ISBN 3-540-50218-1 (Springer)

— This book develops the foundations of a general theory of twisted en­ doscopy: definition of endoscopic groups, study of the correspondance between twisted conjugacy classes and conjugacy classes in endoscopic groups, definition of transfer factors, and finally the stabilization of the elliptic part of the twisted trace formula. The book also develops a theory of duality and Tamagawa numbers for the hypercohomology of complexes T -» U of tori. Résumé (Fondements de l'endoscopie tordue). — Ce livre développe les bases de la théorie générale de l'endoscopie tordue: définitions des groupes endoscopiques, étude de la correspondance entre classes de conjugaison tordue et classes de conjugaison sur un groupe endoscopique, définition du facteur de transfert, enfin stabilisation de la partie elliptique de la formule des traces tordue. Le livre développe également une théorie de dualité et de nombres de Tamagawa pour l'hypercohomologie de complexes T U de tores. © Astérisque 255, SMF 1999

Foundations of twisted endoscopy

Rational conjugacy classes in reductive groups

On the λ-adic representations associated to some simple Shimura varieties

Shimura varieties and twisted orbital integrals

Let G be a connected reductive group over a field F. In this note the author constructs an element e(G) of the Brauer group of F. The square of this element is trivial. For a local field, e(G) may be regarded as an element of { ?I} and is needed for harmonic analysis on reductive groups over that field. For a global field there is a product formula. The purpose of this note is to provide a cohomological interpretation of the sign changes in harmonic analysis on reductive groups over local fields that are caused by inner twistings. The best understood example occurs in the work of Shelstad [8] on inner twistings of real groups. Theorem 6.3 of that paper gives the following character identity: -() = ( 1)q(G')-q(G)X( ) Here G is a connected reductive group over R, G' is its quasi-split inner form, X., and X. are stabilized characters of matched tempered L-packets of G'(R) and G(R), and y' and y are matched regular semisimple elements of G'(R) and G(R). The sign change in the character identity is ()q(G')-q(G) where q(G) is one half of the dimension of the symmetric space attached to G (more precisely, attached to the simply connected cover of the derived group of G). Although q(G) may be only half integral, the difference q(G') q(G) is always integral. It is reasonable to expect an analogous character identity for groups G over a nonarchimedean local field F. We can determine what the sign change must be by considering the Steinberg characters of G(F) and G'(F), since these should be related by the character identity. Let r(G) denote the F-rank of the derived group of G; the Steinberg character of G(F) has value ( I)r(G) on the elliptic regular set of G(F), and hence the sign change in the character identity must be (_l)r(G')-r(G). There are two groups for which the character identity has been proved: the multiplicative group of a central division algebra of dimension d2 over F for d = 2 (see Proposition 15.5 on p. 484 of [4]) and d = 3 (see Theorem 1 of [1]). Later in this paper we will use a cohomological construction to define a sign e(G) = 1 for any connected reductive group G over a local field. Let G' denote a quasi-split inner form of G. We will show that e(G) = (_l)r(G')-r(G) when the base Received by the editors May 10, 1982. 1980 Mathematics Subject Classification. Primary 22E50. Partially supported by the National Science Foundation under Grant MCS 78-02331. ?1983 American Mathematical Society 0002-9947/82/0000-11 93/$03.25

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Robert E. Kottwitz

Papers

Foundations of twisted endoscopy

Rational conjugacy classes in reductive groups

On the λ-adic representations associated to some simple Shimura varieties

Shimura varieties and twisted orbital integrals

Sign changes in harmonic analysis on reductive groups