  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 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 $K_0(\mathcal{A})$ 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 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 $G(\mathbb{F}_q)$.
