Subrepresentations of direct integrals and finite volume homogeneous spaces

Abstract

We prove a result on representations of separable C ∗ {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 G and those of a closed subgroup K K , when G / K G/K is of finite volume. The result is that if an irreducible representation π \pi is contained in ∫ X π x d μ ( x ) \int _X {{\pi _x}} d\mu (x) , then π ⊆ π x \pi \subseteq {\pi _x} for all x x in a set of positive measure. With G G and K K as above, it follows that for each π ∈ G ^ \pi \in \hat G there exists σ ∈ K ^ \sigma \in \hat K with π ⊆ U σ \pi \subseteq {U^\sigma } , the induced representation. Frobenius reciprocity type results are derived as further consequences.