One of the main results of this
paper is a complete description of the spectral decomposition of the quasi-regular
representation of an arbitrary exponential solvable symmetric space. Benoist had
shown previously that such a representation is multiplicity-free, but he was unable to
compute the precise spectrum and spectral measure. More generally, the
quasi-regular representation is considered for any exponential solvable homogeneous
space. In previous work of the author and Messrs. Corwin, Greenleaf and Grélaud,
the analysis of these representations was carried out in the nilpotent case. The
spectral decomposition arrived at was in terms of the Kirillov orbital parameters.
Corresponding results are obtained here for algebraic exponential solvable
homogeneous spaces in case the stability subgroup is either: a Levi component, or its
nilradical is multiplicity-free in the nilradical of the homogeneous group. The
description of the spectral decomposition in the Mackey parameters is also obtained
for these representations.
