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Admissibility For Monomial Representations of Exponential Lie Groups

Let $G$ be a simply connected exponential solvable Lie group, $H$ a closed connected subgroup, and let $τ$ be a representation of $G$ induced from a unitary character $χ_f$ of $H$. The spectrum of $τ$ corresponds via the orbit method to the set $G\cdot A_τ/ G$ of coadjoint orbits that meet the spectral variety $A_τ= f + \h^\perp$. We prove that the spectral measure of $τ$ is absolutely continuous with respect to the Plancherel measure if and only if $H$ acts freely on some point of $A_τ$. As a corollary we show that if $G$ is nonunimodular, then $τ$ has admissible vectors if and only if the preceding orbital condition holds.