The tempered dual of real or p-adic reductive groups, and its noncommutative geometry (joint work with Anne-Marie Aubert)

Abstract: Suppose G is a real or p-adic reductive group. The space of irreducible tempered representations of G comes equipped with the Fell topology, which encodes important phenomena in representation theory. The topology is usefully studied by noncommutative-geometric methods: the tempered dual naturally identifies with the spectrum of the C*-algebra of G, and its connected components identify with the spectra of certain `blocks’ in the C*-algebra.

For real reductive groups, A. Wassermann proved in 1987 that each `block’ has, up to Morita equivalence, a beautiful and simple structure. This was a crucial step in his proof of the Baum-Connes-Kasparov conjecture for G. For p-adic groups, it is not obvious at all that such a structure can exist, but important examples were given by R. Plymen and his students.

In my talk, I will report on joint work with Anne-Marie Aubert which (1) for arbitrary G, gives a geometric condition for the existence of a Wassermann-type structure on a given block, and (2) when G is a quasi-split symplectic, orthogonal or unitary group, explicitly determines the connected components of the tempered dual for which the geometric assumption is satisfied.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

( slides | video )

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Noncommutative Geometry in NYC

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