Multiplicity one for representations corresponding to spherical distribution vectors of class $\rho$

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 $\mathcal{H}_k / \mathcal{G}_k$, with $k$ a locally compact field of characteristic not equal to two. Here $\mathcal{G}$ is a reductive algebraic group defined over $k$ and $\mathcal{H}$ is the fixed point group of an involution $\sigma$ of $\mathcal{G}$ defined over $k$. It is shown then that the natural representation $\mathcal{L}$ of $G_k$ on the Hilbert space $L^2(\mathcal{H}_k /\mathcal{G}_k)$ is multiplicity free if $\mathcal{H}$ is anisotropic. Next a criterion is derived that leads to multiplicity one also in the noncompact situation. Finally, in the nonarchimedean case, a general procedure is given that might lead to showing that a pair $(\mathcal{G}_k,\mathcal{H}_k)$ is a generalized Gelfand pair. Here $\mathcal{G}$ and $\mathcal{H}$ are suitable algebraic groups defined over $k$