Admissibility for Monomial Representations of Exponential Lie Groups

B. Currey   ; V. Oussa

Journal of Lie theory, Tome 22 (2012) no. 2, pp. 481-487

Résumé

Let $G$ be a simply connected exponential solvable Lie group, $H$ a closed connected subgroup, and let $\tau$ be a representation of $G$ induced from a unitary character $\chi_f$ of $H$. The spectrum of $\tau$ corresponds via the orbit method to the set $G\cdot A_\tau / G$ of coadjoint orbits that meet the spectral variety $A_\tau = f + {\frak h}^\perp$. We prove that the spectral measure of $\tau $ is absolutely continuous with respect to the Plancherel measure if and only if $H$ acts freely on some point of $A_\tau$. As a corollary we show that if $G$ is nonunimodular, then $\tau$ has admissible vectors if and only if the preceding orbital condition holds.

Détail

Classification : 22E25, 22E27
Mots-clés : Exponential Lie groups, coadjoint orbits, monomial representations