Journal of Lie Theory 22 (2012), No. 2, 481--487
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
[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
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