ADMISSIBILITY FOR QUASIREGULAR REPRESENTATIONS OF EXPONENTIAL SOLVABLE LIE GROUPS

Let N be a simply connected, connected non-commutative nilpotent Lie group with Lie algebra n of dimension n. Let H be a subgroup of the automorphism group of N. Assume that H is a commutative, simply connected, connected Lie group with Lie algebra h. Furthermore, assume that the linear adjoint action of h on n is diagonalizable with non-purely imaginary eigenvalues. Let tau = Ind(H)(N) (x) (H) 1. We obtain an explicit direct integral decomposition for tau, including a description of the spectrum as a submanifold of (n + h)*, and a formula for the multiplicity function of the unitary irreducible representations occurring in the direct integral. Finally, we completely settle the admissibility question for tau. In fact, we show that if G = N x H is unimodular, then tau is never admissible, and if G is non-unimodular, then tau is admissible if and only if the intersection of H and the center of G is equal to the identity of the group. The motivation of this work is to contribute to the general theory of admissibility, and also to shed some light on the existence of continuous wavelets on non-commutative connected nilpotent Lie groups.