(July 28, 2014) Unitary representations of topological groups Paul Garrett
[email protected] http:=/www.math.umn.edu/egarrett/ We develop basic properties of unitary Hilbert space representations of topological groups. The groups G are locally-compact, Hausdorff, and countably based. This is a slight revision of my handwritten 1992 notes, which were greatly influenced by [Robert 1983]. Nothing here is new, apart from details of presentation. See the brief notes on chronology at the end. One purpose is to isolate techniques and results in representation theory which do not depend upon additional structure of the groups. Much can be done in the representation theory of compact groups without anything more than the compactness. Similarly, the discrete decomposition of L2(ΓnG) for compact quotients ΓnG depends upon nothing more than compactness. Schur orthogonality and inner product relations can be proven for discrete series representations inside regular representations can be discussed without further hypotheses. The purely topological treatment of compact groups shows a degree of commonality between subsequent treatments of Lie groups and of p-adic groups, whose rich details might otherwise obscure the simplicity of some of their properties. We briefly review Haar measure, and prove the some basic things about invariant measures on quotients HnG, where this notation refers to a quotient on the left, consisting of cosets Hg. We briefly consider Gelfand-Pettis integrals for continuous compactly-supported vector valued functions with values in Hilbert spaces. We emphasize discretely occurring representations, neglecting (continuous) Hilbert integrals of representa- tions. Treatment of these discrete series more than suffices for compact groups.