Dimension formula for slice for visible actions on spherical nilpotent orbits in complex simple Lie algebras (Q1625132)

A holomorphic action of a Lie group \(H\) on a connected complex manifold is called strongly visible if there exist a real submanifold \(S\subset D\) and an anti-holomorphic diffeomerphism \(\sigma\) on \(D\) such that \(a)\) \(S\) meets every \(H\)-orbit in \(D\), \(b)\) \(\sigma\mid_S=id_S\), \(c)\) \(\sigma\) preserves each \(H\) orbit in \(D\). Consider a complex finite-dimensional Lie algebra \(g\). Let \(G_{\mathbb C}\) be the group of its inner automorphisms. Take a nilpotent spherical orbit \(O\). Then \(G_{\mathbb C}\) acts on the space \(\mathbb C[O]\), this action is multiplicity-free. The rank of a semigroup of highest weights occurring in \(\mathbb C[O]\) is called the rank of the action of \(G_{\mathbb C}\) on \(O\). The main result of the paper under review is the following theorem. For a strongly visible action of a compact real form of \(G_{\mathbb C}\) on a spherical nilpotent orbit \(O\) one can take a slice such that \(\dim S=\mathrm{rank}_{G_{\mathbb C}}O\). For the entire collection see [Zbl 1392.42001].