Admissibility for Monomial Representations of Exponential Lie Groups

摘要

Let G be a simply connected exponential solvable Lie group, H a closed connected subgroup, and let T be a representation of G induced from a unitary character x_f of H . The spectrum of r corresponds via the orbit method to the set G ? A_T/G of coadjoint orbits that meet the spectral variety f+h~⊥ . We prove that the spectral measure of T is absolutely continuous with respect to the Plancherel measure if and only if H acts freely on some point of A_r . As a corollary we show that if G is nonunimodular, then T has admissible vectors if and only if the preceding orbital condition holds.