Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Lie Theory 22 (2012), No. 2, 481--487Copyright 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 [Abstract-pdf] 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 [ Fulltext-pdf  (268  KB)] for subscribers only.