Gluing of abelian categories and differential operators on the basic affine space

Abstract

The notion of gluing of abelian categories was introduced by Kazhdan and Laumon in an attempt of another geometric construction of representations of finite Chevalley groups; the approach was later developed by Polishchuk and Braverman. We observe that this notion of gluing is a particular case of a general categorical construction (used also by Kontsevich and Rosenberg to define "noncommutative schemes"). We prove a conjecture of Kazhdan which says that the D-module counterpart of the Kazhdan-Laumon gluing construction produces a category equivalent to modules over the ring $\mathcal{D}$ of global differential operators on the basic affine space. As an application we show that $\mathcal{D}$ is Noetherian, and has finite injective dimension as a module over itself.

14 pages

Cited by in corpus (6)

• Introduction to Cluster Algebras. Chapters 1-3
• A new Weyl group action related to the quasi-classical Gelfand-Graev action
• D-modules on rigid analytic spaces III: Weak holonomicity and operations
• Proof of the Ginzburg-Kazhdan conjecture
• Kazhdan-Laumon Category O, Braverman-Kazhdan Schwartz space, and the semi-infinite flag variety
• Polishchuk's conjecture and Kazhdan-Laumon representations