On differential invariants of actions of semisimple Lie groups

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In this paper we suggest an approach to the study of orbits of actions of semisimple Lie groups in their irreducible complex representations. This approach is based on differential invariants on the one hand, and on geometry of reductive homogeneous spaces on the other hand. According to Borel–Weil–Bott theorem, every irreducible representation of semisimple Lie group is isomorphic to the action of this group on the module of holomorphic sections of some one-dimensional bundle over homogeneous space. Using this, we give a complete description of the structure of the field of differential invariants for this action and obtain a criterion, which separates regular orbits.