Differential operators on homogeneous spaces. III: Characteristic varieties of Harish Chandra modules and of primitive ideals. (English) Zbl 0577.22014
[Part I, cf. ibid. 69, 437-476 (1982; Zbl 0504.22015).]
Author’s summary: ”Let G be a semi-simple complex algebraic group with Lie algebra \({\mathfrak g}\) and flag variety \(X=G/B\). For each primitive ideal J with trivial central character in the enveloping algebra U(\({\mathfrak g})\) we define a characteristic variety in the cotangent bundle of X, which projects under the Springer resolution map \(T^*X\to {\mathfrak g}\) onto the closure of a nilpotent orbit. We prove that this characteristic variety is the G-saturation of the characteristic variety of a highest weight module with annihilator J.
”We conjecture that it is irreducible for \(G=SL_ n\). Our conjecture would provide a geometrical explanation for the classification of primitive ideals in terms of Weyl group representations, as achieved by A. Joseph. The presentation of these ideas here is simultaneously used to some extent as an opportunity to continue our more general systematic discussion of differential operators on a complete homogeneous space, and to study more generally characteristic varieties of Harish-Chandra modules”.
Author’s summary: ”Let G be a semi-simple complex algebraic group with Lie algebra \({\mathfrak g}\) and flag variety \(X=G/B\). For each primitive ideal J with trivial central character in the enveloping algebra U(\({\mathfrak g})\) we define a characteristic variety in the cotangent bundle of X, which projects under the Springer resolution map \(T^*X\to {\mathfrak g}\) onto the closure of a nilpotent orbit. We prove that this characteristic variety is the G-saturation of the characteristic variety of a highest weight module with annihilator J.
”We conjecture that it is irreducible for \(G=SL_ n\). Our conjecture would provide a geometrical explanation for the classification of primitive ideals in terms of Weyl group representations, as achieved by A. Joseph. The presentation of these ideas here is simultaneously used to some extent as an opportunity to continue our more general systematic discussion of differential operators on a complete homogeneous space, and to study more generally characteristic varieties of Harish-Chandra modules”.
Reviewer: A.Neagu
MSC:
| 22E46 | Semisimple Lie groups and their representations |
| 14M17 | Homogeneous spaces and generalizations |
| 57T15 | Homology and cohomology of homogeneous spaces of Lie groups |
| 32M10 | Homogeneous complex manifolds |
Keywords:
cohomology groups; Schubert cells; semi-simple complex algebraic group; flag variety; primitive ideal; cotangent bundle; nilpotent orbit; characteristic varieties; Harish-Chandra modulesCitations:
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