The Noncommutative Geometry of Tempered Representations
The Connes-Kasparov isomorphism Nigel HIGSON
Course Description
The purpose of these lectures is to study the tempered dual of a real reductive group as a noncommutative topological space. The unitary dual of a locally compact group may be identified with the spectrum of its group C*-algebra. The C*-algebra point of view equips the unitary dual with a topology, and it also associates to every unitary representation of the group, irreducible or not, a closed subset of the dual. In the case of a real reductive group, the tempered dual is the closed set associated to the regular representation. The tempered dual may also be thought of as the spectrum of the so-called reduced C*-algebra. Following standard practice in C*-algebra theory and noncommutative geometry, we shall interpret the problem of studying the tempered dual as a noncommutative topological space as the problem of studying the reduced C*-algebra up to Morita equivalence.
The extra effort that is required to study the tempered dual in this more elaborate way, and not just a set, is rewarded in spectacular fashion by a beautiful isomorphism statement in K-theory that was conjectured by Connes and Kasparov, and later proved by Wassermann and Lafforgue. I shall describe a proof of the Connes-Kasparov isomorphism for real reductive groups that mostly follows the approach outlined by Wassermann but also uses ideas introduced by Vincent Lafforgue, together with new index-theory calculations that extend Lafforgue’s ideas.
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- Group C*-algebras
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Nigel HIGSON
May. 26, 2017 1:53:27 English
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- Noncommutative topological spaces
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Nigel HIGSON
May. 26, 2017 2:01:18 English
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- Compact and compact modulo center representations
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Nigel HIGSON
May. 26, 2017 1:35:02 English
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- Reductive groups and parabolic subgroups
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Nigel HIGSON
May. 26, 2017 1:51:42 English
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- Introduction to the Connes-Kasparov conjecture
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Nigel HIGSON
May. 26, 2017 1:52:15 English
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- Parabolic induction from a noncommutative-geometric point of view
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Nigel HIGSON
May. 26, 2017 2:01:13 English
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- Determination of the tempered dual as a noncommutative topological space
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Nigel HIGSON
May. 26, 2017 2:05:52 English
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- The Connes-Kasparov isomorphism
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Nigel HIGSON
May. 26, 2017 2:01:13 English
Details
- Year/Term
- 2017 / Intensive, First semester
- Date
- April 14th to May 26th, 2017
- Faculty/
Graduate School - Graduate School of Science
- Language
- English
- Instructor name
- Nigel HIGSON(Distinguished Visiting Professor, Kyoto University / Evan Pugh Professor, Pennsylvania State University)
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