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 $\mathcal{A}$ 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 $G(\mathbb{F}_q)$ . The well-definedness of their construction depended on their conjecture that this category has finite cohomological dimension. This was disproved in 2001 by R. Bezrukavnikov and A. Polishchuk, who found a counterexample in the case $G = SL_3$ . Polishchuk then made an alternative conjecture: though this counterexample shows that the Grothendieck group $K_0(\mathcal{A})$ is not spanned by objects of finite projective dimension, he noted that a graded version of $K_0(\mathcal{A})$ 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 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 $G(\mathbb{F}_q)$ .