Arithmetic harmonic analysis on character and quiver varieties

June 4 to June 8, 2007

at the

American Institute of Mathematics, San Jose, 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: 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.

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

Papers arising from the workshop:

Arithmetic harmonic analysis on character and quiver varieties
by  T. Hausel, E. Letellier, and F. Rodriguez-Villegas
Gale duality and Koszul duality
by  Tom Braden, Anthony Licata, Nicholas Proudfoot, and Ben Webster
Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry
by  David Ben-Zvi, John Francis, and David Nadler