Admissibility for quasiregular representations of exponential solvable Lie groups
Volume 131 / 2013
Abstract
Let $N$ be a simply connected, connected non-commutative nilpotent Lie group with Lie algebra $\mathfrak {n}$ 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 $\mathfrak {h}.$ Furthermore, assume that the linear adjoint action of $\mathfrak {h}$ on $\mathfrak {n}$ is diagonalizable with non-purely imaginary eigenvalues. Let $\tau =\mathop {\rm Ind} _{H}^{N\rtimes H} 1$. We obtain an explicit direct integral decomposition for $\tau $, including a description of the spectrum as a submanifold of $(\mathfrak {n}+\mathfrak {h})^{\ast }$, 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 $G=N\rtimes H$ 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.
