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