Abstract: This talk is about the decomposition of the regular representation of a group like SL(2,R) into its irreducible constituents. The decomposition has both continuous and discrete parts, and more specifically my talk is about the continuous part, which arises through so-called parabolic induction. I'll describe a Hilbert bimodule construction, due to Pierre Clare, that places parabolic induction in a noncommutative-geometric, or C*-algebraic, context. It is also possible to construct an opposite "parabolic restriction" bimodule, although this is not at all trivial. The question of whether the induction and restriction bimodules are adjoint to one another has some interesting geometric aspects. This will be the main focus of the talk.

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