A propos d'un théorème de Vershik et Karpushev
Let \(G\) be a locally compact group and \(\widehat G\) the set of equivalence classes of irreducible unitary representations of \(G\), equipped with Fell's topology. \(G\) is said to have Kazhdan's property \((T)\) if the trivial representation \(1_G\) of \(G\) is an isolated point in \(\widehat G\). Since its introduction this property has proved to be of great importance in several areas of mathematics (see [\textit{P. de la Harpe} and \textit{A. Valette}, Astérisque 175, Soc. Math. de France, 1989; Zbl 0759.22001)]). Regarding representation theory, property \((T)\) turned out to be related to the vanishing of 1-cohomology groups \(H^1(G,\pi)\) of unitary representations \(\pi\) of \(G\). In fact, due to Delorme and Guichardet, \(G\) has property \((T)\) if and only if \(H^1 (G,\pi) =0\) for every unitary representation \(\pi\) of \(G\). For general \(G\), one is led to pay attention to the topology in the neighbourhood of \(1_G\) in \(\widehat G\). More specifically, define the cortex of \(G\), cor \(G\), to be the closed subset of \(\widehat G\) consisting of all points in \(\widehat G\) which cannot be Hausdorff separated from \(1_G\). The first systematic study of cor \(G\) was given in [\textit{M. Bekka} and \textit{E. Kaniuth}, J. Reine Angew. Math. 385, 203-220 (1988; Zbl 0634.22002)]. Now a result of \textit{A. M. Vershik} and \textit{S. I. Karpushev} [Math. USSR, Sbornik 47, 513-526 (1984; Zbl 0528.43005)] says that if \(\pi\) is an irreducible unitary representation of a second countable locally compact group \(G\) such that \(H^1(G,\pi) \neq 0\), then the support of all \(\pi\) (the set of all \(\sigma\in \widehat G\) that are weakly contained in \(\pi)\) is contained in cor \(G\). The main purpose of the paper under review is to extend this latter result to factorial representations. This is achieved by a careful refinement of the proof given by Vershik and Karpushev. Apart from this, the article represents a useful and well-written survey on some aspects of property \((T)\), the cortex and 1-cohomology groups. In particular, various examples are given.
scientific article; zbMATH DE number 1765838