
Journal of Lie Theory 22 (2012), No. 2, 481487 Copyright Heldermann Verlag 2012 Admissibility for Monomial Representations of Exponential Lie Groups Bradley Currey Dept. of Mathematics and Computer Science, 220 N. Grand Blvd., St. Louis, MO 63103, U.S.A. curreybn@slu.edu Vignon Oussa Dept. of Mathematics and Computer Science, 220 N. Grand Blvd., St. Louis, MO 63103, U.S.A. voussa@slu.edu [Abstractpdf] 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. Keywords: Exponential Lie groups, coadjoint orbits, monomial representations. MSC: 22E25, 22E27 [ Fulltextpdf (268 KB)] for subscribers only. 