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

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

2001-04-10

arxiv.org/abs/math/0104114v1

Journ. Inst. Math. Jussieu 1 #4 (2002) 543-557

Mathematics

Representation Theory

14 pages

Scientific paper

10.1017/S1474748002000154

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.

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