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 disproven by R. Bezrukavnikov and A. Polishchuk in 2001, who found a counterexample in the case $G=S{L}_3$. 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 $G({\mathbb{F}}_q)$.

arXiv comment28 pages

References in corpus (4)

• Endoscopy for Hecke categories, character sheaves and representations
• Gluing of abelian categories and differential operators on the basic affine space
• Symplectic Fourier-Deligne transforms on G/U and the algebra of braids and ties
• Kazhdan-Laumon Category O, Braverman-Kazhdan Schwartz space, and the semi-infinite flag variety