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Structure of the center of the algebra of invariant differential operators on certain Riemannian homogeneous spaces. (English) Zbl 1079.43010
The author studies Duflo’s conjecture on the isomorphism between the center of the algebra of invariant differential operators on a homogeneous space and the center of the associated Poisson algebra. For a rather wide class of Riemannian homogeneous spaces, which includes the class of (weakly) commutative spaces, a “weakened version” of this conjecture is proved. It is proved that some localizations of the corresponding centers are isomorphic. For Riemannian homogeneous spaces of the form \(X=(H\times N)/H,\) where \(N\) is a Heisenberg group, Duflo’s conjecture is proved in its original form, i.e. without any localization.
MSC:
43A80 Analysis on other specific Lie groups
53C30 Differential geometry of homogeneous manifolds
17B35 Universal enveloping (super)algebras
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