Admissibility For Quasiregular Representations of Exponential Solvable Lie Groups

TL;DR

This paper provides an explicit decomposition of certain quasiregular representations of exponential solvable Lie groups and characterizes when these representations are admissible based on unimodularity and subgroup intersections.

Contribution

It offers a detailed spectral analysis and criteria for admissibility of quasiregular representations in exponential solvable Lie groups, including explicit formulas and a complete classification.

Findings

Explicit direct integral decomposition of the representation.

Spectral description as a sub-manifold of the dual space.

Admissibility characterized by unimodularity and subgroup intersection conditions.

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, let us assume that the linear adjoint action of $\mathfrak{h}$ on $\mathfrak{n}$ is diagonalizable with non-purely imaginary eigenvalues. Let $\tau=\mathrm{Ind}%_{H}^{N\rtimes H} 1$. We obtain an explicit direct integral decomposition for $\tau$, including a description of the spectrum as a sub-manifold of $(\mathfrak{n}+\mathfrak{h})^{\ast}$, a formula for the multiplicity function of the unitary irreducible representations occurring in the direct integral, and a precise intertwining operator. Finally, we completely settle the admissibility question of $\tau$. In fact, we show that if $G=N\rtimes H$ is unimodular, then $\tau$ is never admissible, and if $G$ is nonunimodular, $\tau$ is admissible if and only if the intersection of $H$ and the center of $G$ is equal to the identity of the group.