Harmonic analysis on exponential solvable homogeneous spaces: The algebraic or symmetric cases (Q1103148)

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 spectrum measure. More generally, the quasi-regular representation is considered for any exponential solvable homogeneous space. In previous work of the author and 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; contained in a normal subgroup; 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.