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GLUING OF ABELIAN CATEGORIES AND DIFFERENTIAL OPERATORS ON THE BASIC AFFINE SPACE

Published online by Cambridge University Press:  14 October 2002

Roman Bezrukavnikov ,
Alexander Braverman  and
Leonid Positselskii
Roman Bezrukavnikov
Affiliation:
Department of Mathematics, University of Chicago, Chicago, IL 60637, USA (roman@math.uchicago.edu)
Alexander Braverman
Affiliation:
Department of Mathematics, Harvard University, 1 Oxford St., Cambridge, MA 02138, USA (braval@math.harvard.edu)
Leonid Positselskii
Affiliation:
Independent Moscow University and IHES, Russian Federation (posic@ihes.fr)
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Abstract

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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.

AMS 2000 Mathematics subject classification: Primary 13N10; 16S32; 17B10; 18C20

Keywords

rings of differential operatorssemisimple Lie algebrasgluing of categories
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 1 , Issue 4 , October 2002 , pp. 543 - 557
Copyright
2002 Cambridge University Press