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Differential operators on homogeneous spaces. II: Relative enveloping algebras. (English) Zbl 0702.22019

[For Part I see Invent. Math. 69, 437-476 (1982; Zbl 0504.22015); III, ibid. 80, 1-68 (1985; Zbl 0577.22014).]
The relative enveloping algebra of an H-principal fibre bundle \(X\to Y\) is defined as the sheaf of algebras U on Y made up by all H-invariant differential operators on X. This concept is applied to the principal fibration \(G\to G/H\) of an algebraic group G by a closed subgroup H. Let \({\mathfrak g}=Lie G\). The left G action gives rise to an algebra homomorphism U(\({\mathfrak g})\to \Gamma (G/H,U)\), and so an ideal J of U(\({\mathfrak h})\) gives rise to a homeomorphism U(\({\mathfrak g})\to \Gamma (G/H,U/J)\). The kernel of the homeomorphism turns out to be the ideal I induced from J in U(\({\mathfrak g})\). The main result gives the precise relation between the associated varieties of J resp. I in \({\mathfrak h}\) resp. \({\mathfrak g}\), in the case H parabolic.
Reviewer: A.Neagu

MSC:

22E60 Lie algebras of Lie groups
17B35 Universal enveloping (super)algebras
53C30 Differential geometry of homogeneous manifolds
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
22E30 Analysis on real and complex Lie groups
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