Actions of Borel Subgroups on Homogeneous Spaces of Reductive Complex Lie Groups and Integrability

Let G be a real reductive Lie group, K its compact subgroup. Let A be the algebra of G-invariant real-analytic functions on T ^*(G/K) (with respect to the Poisson bracket) and let C be the center of A. Denote by 2(G,K) the maximal number of functionally independent functions from A\C. We prove that (G,K) is equal to the codimension (G,K) of maximal dimension orbits of the Borel subgroup BG ^C in the complex algebraic variety G ^C/K ^C. Moreover, if (G,K)=1, then all G-invariant Hamiltonian systems on T ^*(G/K) are integrable in the class of the integrals generated by the symmetry group G. We also discuss related questions in the geometry of the Borel group action.