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

Roman Bezrukavnikov, Alexander Braverman, Leonid Positselskii

Published 2001-04-10Version 1

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.