Multiplicities in the Decomposition of Unitary Representations of Reductive Lie Groups.
Lie Groups, Algebraic Groups and Transformation Groups: A conference in honor of Ernest B. Vinberg's 70th birthday (organized by Herbert Abels, Vladimir Chernousov, Gregory Margulis, Detlev Poguntke, and Katrin Tent). Universität Bielefeld, Germany, 20-24 July 2007.

In a celebrated paper by É. A. Vinberg and B. N. Kimelfeld (1978), a geometric characterization in terms of flag varieties was proved for the multiplicities of finite dimensional irreducible representations occurring in the induced/restricted representations to be not greater than one.

Unlike the finite dimensional case, multiplicities are often infinite in the irreducible decompositions for infinite dimensional representations, even in (seemingly) natural settings. In this talk, I plan to discuss finite multiplicity theorems and multiplicity-free theorems, in particular, for unitary representations as a descendant of Vinberg-Kimelfeld's work.

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