PGL(2)-character varieties and Langlands duality over finite fields

by Tommaso Scognamiglio (Universität Heidelberg)

Thursday 9 Jan 2025, 15:00 → 16:00 Europe/Rome

Aula Seminari (Dipartimento di Matematica)

For a Riemann surface $X$ and a complex reductive group $G$, $G$-character varieties are moduli spaces parametrizing $G$-local systems on $X$. When $G=\mathsf{GL}_n$, 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 $\mathsf{GL}_n(\mathbb{F}_q)$. We expect in general that G-character varieties should be related to $\hat{G}(\mathbb{F}_q)$-representation theory, where $\hat{G}(\mathbb{F}_q)$ is the Langlands dual. After having recalled the results for $G=\mathsf{GL}_n$, I will explain how to generalize some of these results when $G=\mathsf{PGL}_2$. In particular, we will see how to relate $\mathsf{PGL}_2$-character varieties and the representation theory of $\mathsf{SL}_2(\mathbb{F}_q)$.
This is joint work with Emmanuel Letellier.