Mapping of nilpotent orbits under embeddings of real forms of exceptional complex Lie algebras. (Q1811402)

Let \({\mathfrak g}_1\), \({\mathfrak g}_2\) be semisimple real Lie algebras and \({\mathfrak g}_1^c\), \({\mathfrak g}_2^c\) their complexifications. The authors consider Lie algebra monomorphisms \({\phi}:{\mathfrak g}_1\to {\mathfrak g}_2\) between various noncompact real forms \({\mathfrak g}_1\) , \({\mathfrak g}_2\) of complex simple Lie algebras \({\mathfrak g}_1^c\), \({\mathfrak g}_2^c\). For each adjoint nilpotent orbit \(\Omega\) of \({\mathfrak g}_1\), the authors determine the adjoint nilpotent orbit of \({\mathfrak g}_2\) which contains the image \({\phi}({\Omega})\). The adjoint nilpotent orbits of \({\mathfrak g}_1\) and \({\mathfrak g}_2\) are themselves parametrized by using the Kostant-Sekiguchi correspondence. In this paper the authors give an explicit description of the maps \(\mu\) and \(\nu\) for some interesting \(\mathbb Z_2\)-embeddings of real forms of complex Lie algebras. Some good results are given in the form of tables. The authors consider \(\mathbb Z_2\)-embeddings and chain of \(\mathbb Z_2\)-embeddings of a split real form. The necessary details about the enumeration of orbits are also given. The exceptional cases are considered in the Appendix.