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Locally compact Gelfand pairs
February 9th, 2012, Goettingen.
Abstract
Consider pairs of operator algebraic quantum groups \mathbb{G} and \mathbb{H} such that \mathbb{H} is compact and is identified as a quantum subgroup of \mathbb{G}. In this talk we focus on the corepresentations of \mathbb{G} that admit vectors that are invariant under the action of \mathbb{H}. In particular, we look at quantum Gelfand pairs. Using von Neumann algebraic techniques we show how a quantum version of the Plancherel-Godement theorem can be established. This decomposition theorem is made precise for extended SU_q(1,1) with
the circle as subgroup. Peculiar fact is that though that the `extension' of SU_q(1,1) excludes this pair from being a Gelfand pair, this example still incorporates all the Gelfand pair properties one could wish for. |
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