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Differential operators on some nilpotent orbits

Authors: T. Levasseur and J. T. Stafford
Journal: Represent. Theory 3 (1999), 457-473
MSC (1991): Primary 14L30, 16S32, 17B20, 58F06
Published electronically: December 3, 1999
MathSciNet review: 1719509
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Abstract: In recent work, Astashkevich and Brylinski construct some differential operators of Euler degree $-1$ (thus, they lower the degree of polynomials by one) on the coordinate ring $\mathcal {O}(\mathbb {O}_{\min }(\mathfrak {g}))$ of the minimal nilpotent orbit $\mathbb O_{\min }(\mathfrak {g})$ for any classical, complex simple Lie algebra $\mathfrak {g}$. They term these operators “exotic” since there is “(apparently) no geometric or algebraic theory that explains them”. In this paper, we provide just such an algebraic theory for ${\mathfrak {sl}}(n)$ by giving a complete description of the ring of differential operators on $\mathbb O_{\min }({\mathfrak {sl}}(n)).$ The method of proof also works for various related varieties, notably for the Lagrangian submanifolds of the minimal orbit of classical Lie algebras for which Kostant and Brylinski have constructed exotic differential operators.

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Additional Information

T. Levasseur
Affiliation: Département de Mathématiques, Université de Brest, 29285 Brest, France

J. T. Stafford
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109

Keywords: Reductive Lie algebras, nilpotent orbits, differential operators, geometric quantization
Received by editor(s): September 7, 1999
Received by editor(s) in revised form: October 13, 1999
Published electronically: December 3, 1999
Additional Notes: The research of both authors was supported in part by NSF grant NSF-G-DMS 9801148
Article copyright: © Copyright 1999 American Mathematical Society