Continuous cohomology and a conjecture of Serre's

Abstract

W. Casselman* (Vancouver) and D. Wigner (Ann Arbor) 0. Let G be the group of ~p-rational points on a connected semi- simple 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 cohomol- ogy 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 cohomol- ogy of a p-adic semi-simple group.