| Summary: | In 1988, D. Kazhdan and G. Laumon constructed the Kazhdan-Laumon category, an abelian category A 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 subscript q). The welldefinedness of their construction depended on their conjecture that this category has finite cohomological dimension. This was disproven by R. Bezrukavnikov and A. Polishchuk in 2001, who found a counterexample for G = SL₃. Since the early 2000s, there has been little activity in the study of Kazhdan-Laumon categories, despite them being beautiful objects with many interesting properties related to the representation theory of G and the geometry of the basic affine space G/U. In the first part of this thesis, we conduct an in-depth study of Kazhdan-Laumon categories from a modern perspective. We first define and study an analogue of the Bernstein-Gelfand-Gelfand Category O for Kazhdan-Laumon categories and study its combinatorics, establishing connections to Braverman-Kazhdan’s Schwartz space on the basic affine space and the semi-infinite flag variety. We then study the braid group action on D superscript b (G/U) (the main ingredient in Kazhdan and Laumon’s construction) and show that it categorifies the algebra of braids and ties, an algebra previously studied in knot theory; we then use this to provide conceptual and geometric proofs of new results about this algebra. After Bezrukavnikov and Polishchuk’s counterexample to Kazhdan and Laumon’s original conjecture, Polishchuk made an alternative conjecture: though the counterexample shows that the Grothendieck group K₀(A) is not spanned by objects of finite projective dimension, he noted that a graded version of K₀(A) can be thought of as a module over Laurent polynomials and conjectured that a certain localization of this module is generated by objects of finite projective dimension. He suggested that this conjecture could lead toward an alternate proof that Kazhdan and Laumon’s construction is well-defined, and he proved this conjecture in Types A₁,A₂,A₃, and B₂. We prove Polishchuk’s conjecture for all types and prove that Kazhdan and Laumon’s construction is indeed well-defined, giving a new geometric construction of discrete series representations of G(F subscript q).
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