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3 - Dirac Operators and Representation Theory

Published online by Cambridge University Press:  06 November 2024

Edited by
Roger Plymen  and
Mehmet Haluk Şengün
Roger Plymen
Affiliation:
University of Manchester
Mehmet Haluk Şengün
Affiliation:
University of Sheffield
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Summary

This chapter is an introduction to the geometric perspective of tempered representations of connected reductive Lie groups via Dirac induction and the Connes–Kasparov isomorphism. We start with discussing Dirac operators in spin geometry. We then recall some classical Dirac operators on homogeneous spaces arising in representation theory. Next, we introduce the theory of higher index to be used as a language to formulate the Connes–Kasparov conjecture. In Section 3.4, we introduce Dirac induction, the Connes–Kasparov conjecture, and survey a couple of ways for proving it. The final section is devoted to the harmonic-analytic approach to the Connes–Kasparov isomorphism illustrated by the example of connected semisimple groups.

Keywords

Dirac-type operatorsFredholm indexgroup C*-algebrasoperator K-theoryhigher indicesConnes–Kasparov isomorphismDirac inductionnoncommutative Fourier analysis
Type
Chapter
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K-Theory and Representation Theory , pp. 99 - 156
Publisher: Cambridge University Press
Print publication year: 2024

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