| Résume | For a Riemann surface X and a complex reductive group G, G-character
stacks are moduli spaces parametrizing G-local systems on X. When
G=GLn,these spaces 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 of the latter results is that the geometry of these
spaces is related to the representation theory of the finite group
GLn(Fq).
In the first part of the talk, I will review some of these results
regarding GLn. In the second part, I will explain how to generalize some
of these results when G=PGL2. In particular, we will study the geometry
of PGL2-character stacks and relate it to the representation theory of
SL2(Fq). This seems to suggest in general that G-character stacks should
be related to the representation theory of G*(Fq), where G* is the
(split) Langlands dual over Fq.
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