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Continuous cohomology and a conjecture of Serre's

1974; Springer Science+Business Media; Volume: 25; Issue: 3-4 Linguagem: Inglês

10.1007/bf01389727

1432-1297

W. Casselman, David Wigner,

Algebraic structures and combinatorial models

Vancouver) and D. Wigner (Ann Arbor) 0. Let G be the group of ~p-rational points on a connected semisimple group defined over ~v, ff its Lie algebra, H* (G+ Qp) the continuous cohomology of G with coefficients in Qp.When G is compact, a result of Lazard's ([7], Chapter V, Theot'em 2.4.10) and an argument about Zariski-closure (see w 3) imply that H* (G, (I)p)~ H* (fr Qp).The original motivation for most of the results in this paper was the question asked by Serre ([II], p. 119): Does Lazard's result hold for more general G? We show this to be so (Theorem 1 in w 3).We include a largely self-contained exposition of continuous cohomology theory for locally compact groups.Our main result here is a form of Shapiro's Lemma (Propositions3 and 4 in w 1).We also include a discussion of the Hochschild-Serre spectral sequence (in w 2).We have drawn largely on a paper of Hochschild and Mostow [-4] which treats the case of G-modules which are real vector spaces, but our emphasis is quite different.In many cases our cohomology agrees with that constructed by Calvin Moore (described in [10]), and a number of our results are implied by results of his.In w 3 we apply Shapiro's Lemma and the Bruhat-Tits building to prove Serre's conjecture.In w 4 we deal with the cohomology of p-adic groups with coefficients in real vector spaces, including some remarks about the cohomology of smooth representations over more general fields.In w 5 we answer a question of Serre's about the analytic cohomology of a p-adic semi-simple group.We wish to thank A. Bore1 for several valuable suggestions; P. Cartier for remarks concerning w and R. Bott for explaining to one of us a long time ago how a form of Shapiro's Lemma offered a simple proof of Theorem 6.1 in [43.(It was this explanation which ultimately suggested the proof of our Theorem 1+) Finally, we wish to thank the referee for the great deal of patience involved in reading and correcting several versions of this paper, and more particularly for suggesting the proof of Hochschild-Serre that we give.1, Let G be a locally compact topological group.Define a G-space to be a topological Hausdorff space A together with a jointly continuous