Journal of Lie Theory

no. 2
Admissibility for Monomial Representations of Exponential Lie Groups
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 
@article{JOLT_2012_22_2_a5,
     author = {B. Currey and V. Oussa},
     title = {Admissibility for {Monomial} {Representations} of {Exponential} {Lie} {Groups}},
     journal = {Journal of Lie Theory},
     pages = {481--487},
     year = {2012},
     volume = {22},
     number = {2},
     doi = {10.5802/jolt.676},
     zbl = {1272.22005},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.676/}
}
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%A V. Oussa
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B. Currey; V. 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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