Arithmetic harmonic analysis on character and quiver varieties

by Emmanuel Letellier, Laboratoire Nicolas Oresme, Université de Caen.

Let S be a k-punctured Riemann surface of genus g, and let C_1, ..., C_k be generic conjugacy classes of GL(n,C). Then one can consider the space M of isomorphism classes of representations \pi_1(S ) ---> GL(n,C) where a simple loop encircling the puncture a_i is sent to \bar C_i.
We are interested in the connection between the geometry of M (mixed Hodge structures), Macdonald symmetric functions and the representation theory of GL_n(Fq) (e.g. multiplicities in tensor products of characters). We will in particular discuss the case k = g = 1.