PGL2-character varieties and Langlands duality over finite fields

Algebraic Geometry and Number Theory Seminar

For a Riemann surface X and a complex reductive group G, G-character varieties are moduli spaces parametrizing G-local systems on X. When G=GLn, the cohomology of these character varieties have been deeply studied and under the so-called genericity assumptions, their cohomology admits an almost full description, due to Hausel, Letellier, Rodriguez-Villegas and Mellit. An interesting aspect is that the geometry of these varieties is related to the representation theory of the finite group GLn(Fq).
We expect in general that G-character varieties should be related to (Fq)-representation theory, where (Fq) is the Langlands dual.
In the first part of the talk, I will recall the results concerning GLn.
In the second part, I will explain how to generalize some of these results when G=PGL2. In particular, we will see how to relate PGL2-character varieties and the representation theory of SL2(Fq). This is joint work with Emmanuel Letellier.