Sur la Propriété (T) tordue par un produit tensoriel

Abstract

In this article, we consider tensor products of unitary representations by irreducible non-unitary finite dimensional representations of topological groups to define a property that is a twisting of Kazhdan's Property (T). We use the uniform decay of the matrix coefficients of unitary representations, to show that for most of the real semi-simple Lie groups having Kazhdan's Property (T), any finite dimensional irreducible representation $ρ$ of $G$, is isolated among representations of the form $ρ\otimesπ$, where $π$ ranges over the irreducible unitary representations, in a sense to be made precise.