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Gluing of Abelian categories and differential operators on the basic affine space. (English) Zbl 1044.16020

Summary: The notion of gluing of Abelian categories was introduced in a paper by Kazhdan and Laumon in 1988 and studied further by Polishchuk. We observe that this notion is a particular case of a general categorical construction.
We then apply this general notion to the study of the ring of global differential operators \(\mathcal D\) on the basic affine space \(G/U\) (here \(G\) is a semi-simple simply connected algebraic group over \(\mathbb{C}\) and \(U\subset G\) is a maximal unipotent subgroup). We show that the category of \(\mathcal D\)-modules is glued from \(|W|\) copies of the category of \(D\)-modules on \(G/U\) where \(W\) is the Weyl group, and the Fourier transform is used to define the gluing data. As an application we prove that the algebra \(\mathcal D\) is Noetherian, and get some information on its homological properties.

MSC:

16S32 Rings of differential operators (associative algebraic aspects)
14M17 Homogeneous spaces and generalizations
18C20 Eilenberg-Moore and Kleisli constructions for monads
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