Degenerate orbits of adjoint representation of orthogonal and unitary groups regarded as algebraic submanifolds

Abstract

We suggest a method for describing some types of degenerate orbits of orthogonal and unitary groups in the corresponding Lie algebras as level surfaces of a special collection of polynomial functions. This method allows one to describe orbits of the types SO(2n)/SO(2k)×SO(2) ^n−k , SO(2n+1)/SO(2k+1)×SO(2) ^n−k , and (S)U(n)/(S)(U(2k)×U(2) ^n−k ) in so(2n), so(2n+1), and (s)u(n), respectively. In addition, we show that the orbits of minimal dimensions of the groups under consideration can be described in the corresponding algebras as intersections of quadries. In particular, this approach is used for describing the orbit CP ⁿ⁻¹⊂u(n)