Morse linear functionals on orbits of adjoint action of semisimple Lie groups

Linear functionals are considered on the Lie algebra of an arbitrary semisimple compact Lie group with restrictions of these functionals onto an arbitrary orbit of the adjoint action. Criteria for the criticality and non-degenerate criticality of a point on the orbit are formulated and proved for a given functional, the necessary and sufficient condition for a linear functional to be a Morse function on the orbit is also proved. The method calculating the indices of critical points and its applications in the study of topological properties of orbits are indicated.