Let $G$ be a locally compact, unimodular group and $Z$ be its center
Clearly, square integrable representations with central unitary character is unitarizabile, since their matrix coeffecient imbed into $L^2(G/Z)$.
What about tempered distributions with unitary central character, whose matrix coefficients embed only into $L^{2+\epsilon}(G/Z)$ for all $\epsilon>0$?