Invariants of algebraic group actions from differential point of view

Abstract

In this paper we suggest an approach to study actions of semisimple (or reductive) algebraic groups in their algebraic complex representations. We use differential-geometric methods instead of classical algebraic constructions. Namely, according to Borel–Weil–Bott theorem, every algebraic representation of semisimple algebraic group is isomorphic to the action of this group on the module of holomorphic sections of some reductive bundle over homogeneous space. Using this, we give a complete description of the field of differential invariants for this action and obtain a criterion, which separates regular orbits.