Complexity of nilpotent orbits and the Kostant-Sekiguchi correspondence (Q2571059)

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 \(\mathcal{O}.\)