On the continuous (co)homology of locally profinite groups and the Künneth theorem
Let \(G\) be a locally compact and totally disconnected group, which we assume to be separated, metrizable, and countable at infinity. The neighbourhood basis of the neutral element \(1_G\) of \(G\) is formed by compact open subgroups, i.e., \(G\) is a locally profinite group. The purpose of this paper is to develop some aspects of a (co)homology (smooth and continuous) of the group \(G\) in a topological locally convex quasi-complete separated complex vector space \(V\). We study two categories: the category \({\mathcal C}_G\) of continuous \(G\)-modules and the category \(\text{Alg}(G)\) of smooth \(G\)-modules, i.e., each vector of the representation space is invariant by an open compact subgroup. The relation between smooth and continuous (co)homologies of the group \(G\) and those of its closed normal subgroup \(H\) is coded in the notion of spectral sequence, which was discovered by G. Hochschild and J.-P. Serre for discrete groups and is extended in this paper to locally compact totally disconnected groups. One of the main difficulties is the fact that it is no longer evident (as for discrete or profinite groups) that the Hecke algebra \(C^\infty_c(G)\) of locally constant functions with compact supports is injective as an \(H\)-module. One might hope that any injective \(G\)-module in \(\text{Alg}(G)\) is \(H\)-acyclic, which guarantees the convergence of the analogue of the Hochschild-Serre spectral sequence. For \(H\) a \(p\)-adic subgroup of \(G\), this is done in [\textit{A. Borel} and \textit{N. Wallach}, Continuous cohomology, discrete subgroups, and representations of reductive groups, Ann. Math. Stud., No. 94 (1980; Zbl 0443.22010), p. 315]. Our result is based on a new method using the fact that the space \(\text{Hom}_{\mathbb{C}}(C^\infty_c(G);V)_\infty\), endowed with the action \(\sigma\) of \(G\) defined by \((\sigma(x)\varphi)(g)=\pi(x\mid\varphi(x^{-1}g))\), is injective as a smooth \(G\)-module, where \((\pi,V)\in\text{Alg}(G)\). This research generalizes results of \textit{J.-P. Serre} [Cohomologie Galoisienne, Lect. Notes Math. 5 (1973; Zbl 0259.12011), p. 14] and of \textit{W. Casselman} [J. Fac. Sci., Univ. Tokyo, Sect. I A 28, 907-928 (1981; Zbl 0519.22011), p. 922] -- stated without proof -- and allows us to prove the Künneth theorem for the smooth cohomology of the product.
scientific article; zbMATH DE number 554217