Alexandre Afgoustidis: Representations of real reductive groups, of their Cartan motion groups, and the Mackey-Higson bijection.

Abstract

The representation theory of real reductive groups is a deep and classical subject: greatly developed by Harish-Chandra, Langlands and many others between 1950 and 1980, and intensively studied since.

To such a group \(G\), one can attach a ‘Cartan motion group’ \(G_0\): a Lie group with the same dimension as \(G\), but a much simpler algebraic structure (extension of a compact group by an abelian group). The representation theory of \(G_0\) is quite simple: classifying its unitary representations was one of the early successes of Mackey’s theory of induced representations in the late 1940s.

In the early 1970s, Mackey remarked that the parameters for his own classification were similar to Harish-Chandra’s much more subtle parameters for the (at the time ongoing) classification of ‘tempered’ representations of \(G\). He conjectured the existence of a natural one-to-one correspondence between large families of representations of G and \(G_0\).

The idea seems to have been disregarded as overly optimistic at the time. It found a new life in the 1990s, when Alain Connes and Nigel Higson pointed out a connection between Mackey’s proposal and the Baum-Connes-Kasparov conjecture in operator K-theory. Nigel Higson expanded on this idea in 2008 and realized special cases of the Mackey bijection.

My talk will describe a natural one-to-one correspondence between the irreducible tempered representations of \(G\) and the irreducible unitary representations of \(G_0\), and say how some of its properties do imply a new proof of the Baum-Connes-Kasparov ‘conjecture’ for real reductive groups.

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