Singularity of orbits in classical Lie algebras (Q1413655)

The singularity properties of orbits in the Lie algebras of the classical compact connected simple Lie groups \(G\) are investigated. The work is motivated by known results (Ragozin (1972)), and the main aim of the paper is to determine any sum of \(\dim G\) nontrivial orbits, in addition to the cases of the classical Lie algebras which have been considered previously (Hare (1998), Hare, Wilson and Yee (2000), Gupta and Hare (2002)). A geometric approach is developed and applied which is based on the study of the tangent spaces to the orbit. The orbits of minimal dimensions on the groups \(\text{SO}(n)\) and \(\text{Sp}(n)\) are determined. The bases of these tangent spaces are found and elementary properties of the related basic vectors are summarized. It is proved that the \(k\)-fold sum of the tangent spaces of the orbits of minimal dimensions for \(k< \text{rank}(G)\) has dimension less than that of the corresponding Lie algebra. It is concluded that the \(k\)-fold sum of these orbits has measure zero. It is also proved that the \(\text{rank}(G)\)-fold sum of any nontrivial adjoint orbit has positive measure. Thus, as the main result, it is shown that for the above choice of \(k\) there exists a central, continuous measure \(\mu\) on the group \(G\) such that \(\mu^k\) is singular to \(L^1(G)\). It is pointed out that this result is sharp for Lie groups other than those of types \(B_n\) or \(C_3\).