C*-algebras and the noncommutative geometry of real reductive groups

Nigel Higson (Penn State)

Abstract:

The irreducible and usually infinite-dimensional representations of real reductive groups (think SL(n,R)) have been studied and classified by Harish-Chandra, Langlands and friends. The standard approach taken by operator algebraists - to construct a convolution algebra on the group, then study irreducible representations of the algebra - has usually *not* been the route followed in representation theory. Nevertheless there are some interesting possibilities in this direction, as I’ll try to indicate in these lectures.

  • Reductive groups, representations and convolution algebras
  • Parabolic induction
  • Structure of the reduced C*-algebra
  • The Mackey analogy

References:

Texts:

Here are two standard texts on representation theory:

They’re not light reading (this is particularly so for the first) but they’re comprehensive (again, this is particularly so for the first).

Here are two smaller, lighter books that focus on SL(2,R).

  • Howe, Tan - Non-abelian harmonic analysis. [MR1151617]
  • Varadarajan - Harmonic analysis on semi simple Lie groups [MR1071183]

The second gives a good introduction to Harish-Chandra’s methods.

Papers:

A selection of papers related in part to what I’ll have to say. They’re listed alphabetically, but as it happens they’re also listed in rough order of appearance in the lectures.

  • Bernstein - Notes of lectures on Representations of p-adic Groups. [pdf] from Bernstein’s web page.
  • Bernstein, Krötz - Smooth Fréchet globalizations of Harish-Chandra modules [arXiv:0812.1684]  [MR3219530]. Sections 1-3.
  • Clare, Crisp, Higson - Parabolic induction and restriction via C*-algebras and Hilbert modules  [arXiv:1409.8654].
  • Cowling, Haagerup, Howe - Almost L2 matrix coefficients. [MR0946351].
  • Higson - The Mackey analogy and K-theory. [MR2391803].
  • Mackey - On the analogy between semi simple Lie groups and certain related semi-direct product groups. [MR0409726].
  • Penington, Plymen - The Dirac operator and the principal series for complex semi simple Lie groups.  [MR0724030].
  • Vogan - Classifying representations by lowest K-types. [MR0789294].
  • Wassermann - Une démonstration de la conjecture de Connes-Kasparov pour les groupes de Lie linéaires connexes réductifs. [MR0894996].

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