Polishchuk's conjecture and Kazhdan-Laumon representations
Calder Morton-Ferguson
TL;DR
Kazhdan–Laumon's gluing construction builds an abelian category $\mathcal{A}$ of glued perverse sheaves to geometrically realize discrete series for $G(\\mathbb{F}_q)$ but hinges on finite cohomological dimension. Bezrukavnikov–Polishchuk showed this conjecture false in general, prompting Polishchuk’s localization framework: the graded Grothendieck group $K_0(\mathcal{A}_{w,\\mathbb{F}_q})$ admits a $\\mathbb{Z}[v,v^{-1}]$-module structure and becomes generated by finite projective dimension objects after inverting a polynomial $p(v)$. The paper proves Polishchuk’s conjecture in full generality via monodromic perverse sheaves, the monodromic Hecke category, and its center, plus a Barr–Beck–Lurie dg-formalism, to show the localization of $K_0(\mathcal{A}_{w,\\mathbb{F}_q}^{\\mathcal{L}})$ is generated by finite projective objects. It then establishes that Kazhdan–Laumon’s construction is well-defined for monodromic data, enabling a geometric construction of discrete series representations by pairing with a Grothendieck–Lefschetz-type functional and passing to the localized $K_0$–spaces. The approach unifies monodromic and parabolic variants, extends Braverman–Polishchuk’s quasi-regular case, and provides a broad framework for explicit realizations of $G(\\mathbb{F}_q)$-representations from $\\mathcal{A}$. Overall, the work completes Kazhdan–Laumon’s program by proving the well-definedness of their representations in full generality and connecting categorical centers, canonical complexes, and dualities to representation theory of finite groups of Lie type.
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 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 $G(\mathbb{F}_q)$.