Tempered representation

Remarks

The functions $\ip{\pi(g) v}{v}$ are diagonal matrix coefficients of $\pi$. For a second countable unimodular group of type I, the tempered representations form the support of the Plancherel measure.

For a connected semisimple Lie group $G$ with finite centre and maximal compact subgroup $K$, a theorem of Cowling, Haagerup and Howe gives an equivalent form. An irreducible unitary representation $\pi$ is tempered if and only if its $K$-finite matrix coefficients lie in $L^{2 + \varepsilon}(G)$ for every $\varepsilon > 0$. For reductive groups this growth condition is often taken as the definition.

Example: for $G = \R^n$, or any abelian locally compact group, every unitary character is tempered, since $\lambda_G$ weakly contains every character. Example: a discrete series representation of a semisimple group is tempered, since it is a subrepresentation of $\lambda_G$.

Non-example: for $G = \mathrm{SL}(2, \R)$ the trivial representation is not tempered. Its matrix coefficient is the constant function $1$, which is not in $L^{2+\varepsilon}(G)$ because $G$ has infinite Haar measure.