Tempered Representation

A tempered representation of a locally compact group is a unitary representation weakly contained in its regular representation. Weak containment compares its matrix coefficients with those of the regular representation on compact subsets. For an irreducible unitary representation that is admissible, of a reductive group over a local field of characteristic 0, an equivalent criterion uses integrability of the absolute values of its matrix coefficients modulo the center to every exponent , with . Here the coefficients are formed from vectors whose translates under a maximal compact subgroup each span a finite-dimensional vector space (Cowling et al. 1988; Nelson and Venkatesh 2021, §23.4.1). Temperedness expresses a restriction on the size of local spectral parameters.