Multiplicity one for representations corresponding to spherical distribution vectors of class ρ

In this paper one considers a unimodular second countable locally compact group G and the homogeneous space X:=H\G, where H is a closed unimodular subgroup of G. Over X complex vector bundles are considered such that H acts on the fibers by a unitary representation rho with closed image. The natural action of G on the space of square integrable sections is unitary and possesses an integral decomposition in so-called spherical distributions of class rho. The uniqueness of this decomposition can be characterized by a number of equivalent properties. Uniqueness is shown to hold for a class of semidirect products. In the case that H is compact, the multiplicity free decomposition is shown to be equivalent with the commutativity of a suitable convolution algebra. As an example, one takes for X a symmetric k-variety H-k\g(k), with k a locally compact field of characteristic not equal to two, and for rho a character of H-k, whose square is trivial. Here g is a reductive algebraic group defined over k and H is the fixed point group of an involution sigma of g defined over k. It is shown then that the natural representation L of G(k) on the Hilbert space L-2(H-k\g(k)) is multiplicity free if H is anisotropic. Next a criterion is derived that leads to multiplicity one also in the noncompact situation. Finally, in the non-Archimedean case, a general procedure is given that might lead to showing that a pair (g(k),H-k) is a generalized Gelfand pair. Here g and H are suitable algebraic groups defined over k.