Skrypnik T. V.
Ukr. Mat. Zh. - 1999. - 51, № 12. - pp. 1714–1718
We generalize the notion of stereographic projection to the case of an arbitrary compact Lie group and find the explicit form of the local complex parametrization of an orbit of the corresponding group.
Explicit realization of irreducible representations of classical compact lie groups in the spaces of sections of line bundles
Ukr. Mat. Zh. - 1998. - 50, № 10. - pp. 1316–1323
We use the Borel-Weil scheme for the construction of irreducible representations of compact Lie groups in the spaces of holomorphic sections of line bundles over homogeneous manifolds. We find the explicit form of the space of sections and construct an invariant scalar product. We show that the space of holomorphic sections locally satisfies the Zhelobenko indicator system.
Degenerate orbits of adjoint representation of orthogonal and unitary groups regarded as algebraic submanifolds
Ukr. Mat. Zh. - 1997. - 49, № 7. - pp. 895–905
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).