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 hyperkähler moduli spaces, which arise by a finite holomorphic symplectic quotient construction.
These include: toric hyperkähler varieties, Nakajima's quiver varieties, Hilbert schemes of n points and moduli spaces of Yang-Mills instantons on C^2;
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.