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Invariant linear connections on homogeneous symplectic varieties. (English) Zbl 1009.22014

Let \(G\) be a connected Lie group. One can consider the following problem: describe all homogeneous \(G\)-spaces \(X\) that admit an invariant linear connection. Not much is known in general and the main attention was focused on the description of invariant linear connections on some particularly nice homogeneous spaces, mainly on reductive homogeneous spaces. The authors give a complete solution of this problem in the following situation: \(G\) is a connected semisimple Lie group over the fields \(\mathbb R\) or \(\mathbb C\) and \(X\cong G/H\) is a symplectic \(G\)-variety, that is, \(X\) is either an adjoint orbit of \(G\) or its covering.

MSC:

22E46 Semisimple Lie groups and their representations
53C30 Differential geometry of homogeneous manifolds
53D35 Global theory of symplectic and contact manifolds
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