Arithmetic geometry of representation spaces

Tuesday, 14-November-2023 

Abstract

The moduli space of representations of the fundamental group of a Riemann surface is a central object in mathematics with deep connections to Higgs bundles, flat connections, and Yang—Mills theory. Huge amount of research is done on these moduli spaces "in type A" i.e., for representations into the group GL_n. However, if we consider representations into groups of more general type (required for applications to Langlands duality and mirror symmetry), then very little is known about the geometry of representation spaces (e.g. Betti numbers are not known). I will discuss how one can utilise representation theory of finite groups of Lie type (a la Deligne—Lusztig) to get a handle on these spaces and compute some of their invariants.