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  • 3 - Dirac Operators and Representation Theory

  • Edited by Roger Plymen, University of Manchester, Mehmet Haluk Şengün, University of Sheffield
  • Book:
    K-Theory and Representation Theory
    Published online:
    06 November 2024
    Print publication:
    28 November 2024, pp 99-156

  • 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.