#
Arithmetic harmonic analysis on character and quiver varieties

June 4 to June 8, 2007
at the

American Institute of Mathematics,
Palo Alto, California

organized by

Tamas Hausel,
Emmanuel Letellier,
and Fernando Rodriguez-Villegas

## Original Announcement

This workshop will be devoted
to bringing together mathematicians
working on the following circle of ideas:
- cohomology of character and quiver varieties
- representation theory of finite groups and algebras of Lie type
- applications of the Weil conjectures to cohomological calculations
- geometric representation theory of various finite and infinite
dimensional algebras
- combinatorics of Macdonald polynomials

Geometrical methods, pioneered by Borel-Weil-Bott, Deligne-Lusztig,
Kazhdan-Lusztig, Ginzburg, Nakajima etc play a central role in
representation theory. The idea is to study representations of various
algebraic objects on the cohomology of various varieties. Many of the
varieties appearing are examples of Nakajima's quiver varieties.
Star-shaped quiver varieties also appear in the non-Abelian Hodge
theory of a Riemann surface. Via the Riemann-Hilbert monodromy map
they are related to the character variety which is the representation
variety of the fundamental group of the Riemann surface to a complex
reductive Lie group. The Riemann-Hilbert map in turn relates the
cohomologies of the varieties in an intriguing way.

However until recently the cohomology of character varieties have not
been studied from the perspective of representation theory. Recently
it was found that arithmetic methods could be used to study their
cohomology, and in turn relate them to the representation theory of
finite groups of Lie type. The analogue arithmetic study on quiver
varieties leads to the representation theory of finite Lie algebras.
The Riemann-Hilbert monodromy map then conjecturally relates the two
representation theories in a surprising way. In particular,
conjecturally, the cohomology of character varieties is intimately
related to Macdonald polynomials, which are of great interest in
combinatorics and representation theory.

Specifically, we would like to address
the following questions:

- Is there a topological quantum field theory that governs the
geometry of the character varieties in question?
- What are all of the implications of the purity conjecture (relating
the cohomologies of the character varieties and the associated
quiver varieties) for the representations theory of groups of Lie
type and their algebras? Can we prove the purity conjecture?
- Is there a relation between the natural generating series arising
from counting points on the character varieties over finite fields
and modular forms?
- What exactly is the significance and what are the consequences of
the appearance of the Macdonald polynomials in this geometric
setting?

Overall, our hope is that the workshop will be an opportunity for
fruitful interactions between the different research areas involved.

## Material from the workshop

A list of participants.
The workshop schedule.

A report on the workshop activities.

Scans of lectures notes of workshop talks.

Fernando Rodriguez-Villegas has a blog for the workshop followup