# Kazhdan-Laumon categories and representations of G(F_q)

In 1988, Kazhdan and Laumon constructed an abelian category associated to a reductive group G over a finite field, with the aim of using it to construct discrete series representations of the finite Chevalley group G(F_q). The well-definedness of their construction depended on their conjecture that this category has finite cohomological dimension. This was disproven by Bezrukavnikov and Polishchuk in 2001. Subsequently, Polishchuk made an alternative conjecture, that a certain localization of the Grothendieck group of any Kazhdan-Laumon category is generated by objects of finite projective dimension. He suggested that this conjecture was still enough to prove that Kazhdan and Laumon's construction is well-defined, and he proved this conjecture in Types A1, A2, A3, and B2. In this talk I will explain an overview of my results about the structure and symmetry of Kazhdan-Laumon categories, ultimately leading up to a proof of Polishchuk’s conjecture in all types, which validates Kazhdan and Laumon’s construction as a new categorical representation theory construction of representations of finite groups of Lie type.