 If there is a compact group action of a group "G" on the compact manifold "X", commuting with the elliptic operator, then one can replace ordinary K theory in the index theorem with equivariant K-theory.For trivial groups "G" this gives the index theorem, and for a finite group "G" acting with isolated fixed points it gives the Atiyah–Bott fixed point theorem.In general it gives the index as a sum over fixed point submanifolds of the group "G".