/pdf/the-geometry-and-cohomology-of-some-simple-shimura-varieties-4tssmg8muy.pdf

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)

(1.1) This paper concerns three aspects of the action of a compact group K on a space X . The ®rst is concrete and the others are rather abstract. (1) Equivariantly formal spaces. These have the property that their cohomology may be computed from the structure of the zero and one dimensional orbits of the action of a maximal torus in K. (2) Koszul duality. This enables one to translate facts about equivariant cohomology into facts about its ordinary cohomology, and back. (3) Equivariant derived category. Many of the results in this paper apply not only to equivariant cohomology, but also to equivariant intersection cohomology. The equivariant derived category provides a framework in both of these may be considered simultaneously, as examples of ``equivariant sheaves''. We treat singular spaces on an equal footing with nonsingular ones. Along the way, we give a description of equivariant homology and equivariant intersection homology in terms of equivariant geometric cycles. Most of the themes in this paper have been considered by other authors in some context. In Sect. 1.7 we sketch the precursors that we know about. For most of the constructions in this paper, we consider an action of a compact connected Lie group K on a space X , however for the purposes of the introduction we will take K  S1 to be a torus. Invent. math. 131, 25±83 (1998)

/pdf/equivariant-cohomology-koszul-duality-and-the-localization-2973jtaovy.pdf

Equivariant cohomology, Koszul duality, and the localization theorem

Let F be a P-adic field, let L be the completion of a maximal unramified extension of F, and let σ be the Frobenius automorphism of L over F. For any connected reductive group G over F one denotes by B(G) the set of σ-conjugacy classes in G(L) (elements x,y in G(L) are said to be σ-conjugate if there exists g in G(L) such that g-1κ σ(g)=y. One of the main results of this paper is a concrete description of the set B(G) (previously this was known only in the quasi-split case).

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Isocrystals with additional structure. II

The Hasse-Weil zeta functions of varieties over number fields are conjecturally products (and quotients) of automorphic L-functions. For a Shimura variety S associated to a connected reductive group Gover Q one can hope to be more specific about which automorphic L-functions appear in the zeta function. In fact Eichler, Shimura, Kuga, Sato, and Ihara, who studied GL2 and its inner forms, found in those cases that it was enough to use automorphic L-functions for the group G itself. In the general case Langlands [L2-L4] has given a conjectural description of the zeta function in terms of automorphic L-functions for G and its endoscopic groups [LS] (see also [KS] for the contribution of non-tempered representations), and a description of this type has been verified in certain cases, beginning with [L3]. For GL2 it was possible to use the Eichler-Shimura congruence relation in order to make the connection between the zeta function and automorphic Lfunctions. In general one needs more information than the Eichler-Shimura congruence relation gives, and it seems to be necessary to describe the points on S over finite fields in terms of group-theoretical data (for the group G), in a way that is adapted to an eventual comparison of the number of points modulo p with the Selberg trace formula for G (actually with the stable trace formulas for G and its endoscopic groups), as is explained in [L2, L3, KS]. Ihara [11, 12] gave such a group-theoretical description of points modulo p in the case of GL2(Q) and its inner forms, and Deligne [D!] gave a related description of the category of ordinary abelian varieties over a finite field. Langlands [Ll] conjectured a group-theoretical description in the general case, based on a detailed though incomplete study of Shimura varieties of PEL type [S]; Milne [Mil] gave a simplified exposition of this work of Langlands in a special case, using the description of the category of abelian varieties up to isogeny over a finite field due to Honda [H] and Tate [T2, T3]. Zink [Z2] gave complete proofs for part of Langlands's conjectures for Shimura varieties of PEL type, but by that time it was clear that a new idea was needed to give a complete proof for the full conjecture, even for the case of the group of symplectic similitudes. In fact the conjecture itself needed some refinement; this was one of the objects of some work by Langlands-Rapoport [LR], whose main goal, however, was to put the conjecture into a Tannakian framework, in which it became conceptually clearer. The final step was taken independently by

https://www.ams.org/jams/1992-05-02/S0894-0347-1992-1124982-1/S0894-0347-1992-1124982-1.pdf

Points on some Shimura varieties over finite fields

Consider a connected reductive group G over a number field F. For technical reasons we assume that the derived group of G is simply connected (see [L1]). in [L3] Langlands partially stabilizes the trace formula for G. After making certain assumptions, he writes the elliptic regular part of the trace formula for G as a linear combination of the elliptic G-regular parts of the stable trace formulas for the elliptic endoscopic groups H of G. The function f/ used in the stable trace formula forH is obtained from the function f used in the trace formula for G by transferring orbital integrals.

/pdf/stable-trace-formula-cuspidal-tempered-terms-1kchlchcsl.pdf

Robert E. Kottwitz

Papers

Equivariant cohomology, Koszul duality, and the localization theorem

Isocrystals with additional structure. II

Points on some Shimura varieties over finite fields

Stable trace formula: cuspidal tempered terms

Stable trace formula: Elliptic singular terms