The Noncommutative Geometry of Tempered Representations
Group C*algebras 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 socalled 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 Ktheory that was conjectured by Connes and Kasparov, and later proved by Wassermann and Lafforgue. I shall describe a proof of the ConnesKasparov isomorphism for real reductive groups that mostly follows the approach outlined by Wassermann but also uses ideas introduced by Vincent Lafforgue, together with new indextheory calculations that extend Lafforgue’s ideas.

 Group C*algebras

Nigel HIGSON
May. 26, 2017 1:53:27 English

 Noncommutative topological spaces

Nigel HIGSON
May. 26, 2017 2:01:18 English

 Compact and compact modulo center representations

Nigel HIGSON
May. 26, 2017 1:35:02 English

 Reductive groups and parabolic subgroups

Nigel HIGSON
May. 26, 2017 1:51:42 English

 Introduction to the ConnesKasparov conjecture

Nigel HIGSON
May. 26, 2017 1:52:15 English

 Parabolic induction from a noncommutativegeometric point of view

Nigel HIGSON
May. 26, 2017 2:01:13 English

 Determination of the tempered dual as a noncommutative topological space

Nigel HIGSON
May. 26, 2017 2:05:52 English

 The ConnesKasparov isomorphism

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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