Polishchuk’s conjecture and Kazhdan-Laumon representations

Abstract

In their 1988 paper "Gluing of perverse sheaves and discrete series representations," D. Kazhdan and G. Laumon constructed 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_q). The well-definedness 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 in the case G = SL_3. In the same paper, Polishchuk then made an alternative conjecture: though this counterexample shows that the Grothendieck group K_0(A)$ is not spanned by objects of finite projective dimension, he noted that a graded version of K_0(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, and suggested that this conjecture could lead toward a proof that Kazhdan and Laumon’s construction is well-defined. He proved this conjecture in Types A_1, A_2, A_3, and B_2. In the present paper, we prove Polishchuk’s conjecture in full generality, and go on to prove that Kazhdan and Laumon’s construction is indeed well-defined, giving a new geometric construction of discrete series representations of G(F_q).