Arithmetic Geometry of Character Varieties

Character variety of surface groups plays a central role in diverse areas of mathematics such as Geometric Langlands program and non-abelian Hodge theory. Determining cohomology of the character variety has been a subject active research for decades. In this talk, I will report an on-going project to count points on character varieties over finite fields. The main goal is to generalise the work of Hausel—Letellier-Villegas from type A to arbitrary type. A key role is played by representation theory of finite reductive groups (Deligne—Lusztig theory, Lusztig’s Jordan decomposition, etc.)