Character varieties for Riemann surfaces and representation theory of finite reductive groups
Data: 24 GIUGNO 2025 dalle 11:15 alle 13:00
Luogo: aula Vitali, ore 11:15
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 is well understood. Under the so-called genericity assumptions, (i.e. in the smooth case), their cohomology admits an almost complete description, due to Hausel, Letellier, Rodriguez-Villegas and Mellit. An interesting aspect is that the geometry of these varieties turns out to be related to the representation theory of the finite group GLn(Fq).
We expect in general that G-character varieties should be related to the representation theory of Ĝ(Fq), where Ĝ(Fq) is the Langlands dual.
In the first part of the talk, I will recall the background and the results concerning GLn. Then, 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.