Tempered representation

Summary

A tempered representation is an irreducible unitary representation of a locally compact group that is weakly contained in the left regular representation. For a semisimple Lie group this means that its matrix coefficients are almost square integrable. For a unimodular group of type I, the tempered representations are those needed to decompose $L^2(G)$. They form the support of the Plancherel measure. In the Langlands classification every irreducible admissible representation of a real reductive group is built from tempered ones.