Séminaire Algèbre Géométrie Algébrique Topologie Algébrique :

Le 20 octobre 2005 à 11:00 - salle 431

Présentée par Hausel Tamas - Univ. Oxford / Univ. of Texas at Austin

Arithmetic harmonic analysis, Macdonald polynomials and the topology of the Riemann-Hilbert monodromy map

We show that abelian and non-abelian Fourier transform over finite fields is the right tool to count solutions of holomorphic moment map equations over finite fields. Using the character theory of $GL(n,F_q)$, due to Green and of $gl(n,F_q)$ due to Letellier, this will give a wealth of information on Betti numbers of those hyperkahler moduli spaces, which arise by a finite holomorphic symplectic quotient construction. These include: toric hyperkahler varieties, Nakajima's quiver varieties, Hilbert schemes of n points and moduli spaces of Yang-Mills instantons on C2; $GL (n,C)$ representation varieties of Riemann surfaces, and moduli spaces of flat $GL(n,C)$ connections on algebraic curves. This is partly joint work with Emmanuel Letellier and Fernando Rodriguez-Villegas.