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# Geometric Quantization and Operator K-Theory

### Jeff Fox

University of Colorado

###
December 3, 1998

102 Bradley Hall, 4 pm

Tea 3:30 pm, Math Lounge

**Abstract: **
One of the most important ideas in the unitary representation theory
of Lie groups is that of geometric quantization. It was first proposed
by Kostant and then developed by Kirillov to give a complete
description of the unitary dual for nilpotent Lie groups. The scope of
geometric quantization was expanded by a number of different people in
a number of different directions and it is fair to say that geometric
quantization has been a guiding principle in almost all of the work
done in the last 30 years on the representation theory of locally
compact groups.

In this talk we describe the picture that geometric
quantization gives for the simple Lie group $SU(1,1)$. Semisimple Lie
groups have no finite dimensional irreducible unitary representations,
but do have finite dimensional non-unitary representations (the\lq\lq
Unitary Trick\rq\rq of Weyl). These finite dimensional non-unitary
representations control the infinite dimensional unitary
representation theory of semisimple Lie groups in a very concrete way
via the mechanisms of quantization.

If $G$ is a locally compact group, Kasparov introduced the ring of
Fredholm representations, $R(G)$. This ring, or more precisely a
certain idempotent known (sometimes) affectionately as Kasparov's
gamma element,$\gamma \in R(G)$, has been a decisive tool in computing
the K-theory of the reduced $C^{*}$-algebra $C^{*}_{r}(G)$. We will
describe a conjectural correspondence between geometric quantization
and Kasparov's $R(G)$ for semisimple Lie groups. One advantage of this
correspondence, besides giving a fairly natural geometric construction
of elements in $R(G)$, is that-in some cases-it is also true.

This talk will be accessible to graduate students.