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

This workshop, sponsored by AIM and the NSF, 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:

1. Is there a topological quantum field theory that governs the geometry of the character varieties in question?
2. 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?
3. Is there a relation between the natural generating series arising from counting points on the character varieties over finite fields and modular forms?
4. 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.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.