Journal of Lie Theory

no. 2
Admissibility for Monomial Representations of Exponential Lie Groups
Bradley Currey1; Vignon Oussa1
1Dept. of Mathematics and Computer Science, 220 N. Grand Blvd., St. Louis, MO 63103, U.S.A.
Journal of Lie Theory, Volume 22 (2012) no. 2, pp. 481-487
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.
DOI: 10.5802/jolt.676
Classification: 22E25, 22E27
Keywords: Exponential Lie groups, coadjoint orbits, monomial representations

Bradley Currey  1 ; Vignon Oussa  1

1 Dept. of Mathematics and Computer Science, 220 N. Grand Blvd., St. Louis, MO 63103, U.S.A. 
Bradley Currey; Vignon Oussa. Admissibility for Monomial Representations of Exponential Lie Groups. Journal of Lie Theory, Volume 22 (2012) no. 2, pp. 481-487. doi: 10.5802/jolt.676
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