Complexity of nilpotent orbits and the Kostant-Sekiguchi correspondence. (English) Zbl 1077.22018
Let \(G\) be a connected linear semisimple Lie group. The author proves that, if \((\Omega, \mathcal O)\) is a Kostant-Sekiguchi pair, then the corank of the Hamiltonian \(K\)-space is twice the complexity of the \(K_{\mathbb C}\) variety \(\mathcal{O}.\) Among others, a consequence is derived, which could be used for the computation of the rank and the complexity of \(\mathcal{O}\) in certain cases. The main tool used is a result of Vergne, namely that there is a \(K\)-equivariant diffeomorphism which maps \(\Omega\) onto \(\mathcal{O}.\)
Reviewer: Svatopluk Krýsl (Praha)
MSC:
| 22E46 | Semisimple Lie groups and their representations |
| 14R20 | Group actions on affine varieties |
| 53D20 | Momentum maps; symplectic reduction |
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