On tempered representations

Let $G$ be a unimodular locally compact group. We define a property of irreducible unitary $G$-representations $V$ which we call c-temperedness, and which for the trivial $V$ boils down to F{\o}lner's condition (equivalent to the trivial $V$ being tempered, i.e. to $G$ being amenable). The property of c-temperedness is a-priori stronger than the property of temperedness. We conjecture that for semisimple groups over local fields temperedness implies c-temperedness. We check the conjecture for a special class of tempered $V$'s, as well as for all tempered $V$'s in the cases of $G := SL_2 (\mathbb{R})$ and of $G = PGL_2 (\Omega)$ for a non-Archimedean local field $\Omega$ of characteristic $0$ and residual characteristic not $2$. We also establish a weaker form of the conjecture, involving only $K$-finite vectors. In the non-Archimedean case, we give a formula expressing the character of a tempered $V$ as an appropriately-weighted conjugation-average of a matrix coefficient of $V$, generalising a formula of Harish-Chandra from the case when $V$ is square-integrable.