**Nigel Higson
lectures** 17-21 May 2010

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Nigel Higson (

His
first lecture and one by Terry Gannon (Alberta) will comprise a Spitalfields
Day on the afternoon of 17 May.

**Timetable**

**Spitalfields**** Day** (lectures will be held in room
M/0.40):

Monday 13.00-14.00 Buffet lunch (room M/1.04)

14.00 Terry Gannon (

15.45 Nigel Higson
(

**Rest of week**
(lectures will be held in room M/2.06):

Tuesday 14.00 Higson II

Wednesday 14.00 Gannon II

Thursday 14.00 Higson III

Friday 14.00 Higson IV

**Nigel Higson: The
Baum-Connes Conjecture and Group Representations**

Operator algebra K-theory has well-known applications in topology and geometry stemming from the index theory of Dirac operators and the Baum-Connes conjecture. But the same techniques also resonate in various ways with Lie groups and representation theory. In this series of lectures I shall try to indicate how this comes about, focusing on some fairly new aspects of the relationship.

**Lecture 1 (Spitalfields Talk): C*-algebras, unitary group representations
and topology**

*C**-algebras and the
theory of unitary group representations are both roughly sixty years old. The two subjects were practically one and the
same during their first decade, but diverged soon after. I shall sketch some of the early history that
*C**-algebras and then describe
developments coming from geometry and index theory that are reconnecting *C**-algebras to group representations,
particularly in ways that involve the topological structure of the space
irreducible unitary group representations.

**Lecture 2: Contractions
of Lie groups and the Mackey analogy**

Let *K* be the
maximal compact subgroup of a connected Lie group *G*. The
"contraction" of *G* along *K* is the semidirect
product group associated to the adjoint action of *K* on the quotient of the Lie algebras of *G* and *K*. George Mackey proposed that when *G* is semisimple
there ought to be an "analogy" between the unitary representation
theories of *G* and its
contraction. As I shall explain,
Mackey's proposal is very closely related to the Baum-Connes
conjecture for *G*. I shall examine the particular case of
complex semisimple groups, and also briefly discuss
the real case.

**Lecture 3: Harish-Chandra
homomorphisms**

This is a continuation of the previous lecture. I shall look at the Mackey analogy for admissible
rather than unitary representations, using convolution algebras of
distributions on *G* rather than the
group *C**-algebra. From this point of view the analogy amounts
to a certain generalization of the Harish-Chandra isomorphism theorem in Lie
algebra theory.

**Lecture 4: The Weyl character formula in KK-theory**

Weyl's formula describes the characters of the irreducible representations of compact Lie groups. It has a beautiful relationship with K-theory and index theory, as was pointed out by Atiyah and Bott a long time ago. I shall revisit the subject from the perspective of Kasparov's KK-theory. There are interesting links to the Baum-Connes conjecture that in turn suggest interesting links between Baum-Connes and geometric representation theory.

**Participants**

For a list of participants, click here.

**Support**

EU-NCG members should contact their node coordinators about the possibility of funding for attending these lectures.

There are limited funds available to contribute in part to
the expenses of members of the Society or research students registered at

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