Conditionally negative definite functions, irreducible representations and property (T) (Q2565959)

Summary: This paper is devoted to conditionally negative definite functions on a locally compact group \(G\), and their relation to representation theory and 1-cohomology. More precisely, we prove first that a normalized, conditionally negative definite function \(\psi\) on \(G\) is indecomposable if and only if the orthogonal representation of \(G\) constructed by GNS-construction is irreducible. Next, we define conditionally negative definite measures on \(G\) and we prove that a Radon measure \(d\mu\) absolutely continuous with respect to Haar measure \(dx\) is conditionally negative definite if and only if the Radon-Nikodým derivative \({d\mu\over dx}\) is a conditionally negative definite function. We use this to prove that, on a compactly generated group \(G\), any normalized conditionally negative definite function is the limit, uniformly on compact subsets of \(G\), of convex combinations of indecomposable normalized conditionally negative definite functions. As a consequence, if a compactly generated group has the property that the reduced 1-cohomology is zero for every irreducible representation of \(G\), then the same holds for every unitary representation of \(G\). This is related to a characterisation, by \textit{Y. Shalom} [Invent. Math. 141, 1--54 (2000; Zbl 0978.22010)], of property (T) for compactly generated groups.