Subrepresentations of direct integrals and finite volume homogeneous spaces

Abstract:

We prove a result on representations of separable ${C^*}$-algebras which has application to, and was in fact motivated by, a problem concerning relations between unitary representations of a second countable locally compact group $G$ and those of a closed subgroup $K$, when $G/K$ is of finite volume. The result is that if an irreducible representation $\pi$ is contained in $\int _X {{\pi _x}} d\mu (x)$, then $\pi \subseteq {\pi _x}$ for all $x$ in a set of positive measure. With $G$ and $K$ as above, it follows that for each $\pi \in \hat G$ there exists $\sigma \in \hat K$ with $\pi \subseteq {U^\sigma }$, the induced representation. Frobenius reciprocity type results are derived as further consequences.