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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 $\epsilon>0$?

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