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 epsilon (G,K) the maximal number of functionally independent functions from A C. We prove that epsilon (G,K) is equal to the codimension delta (G,K) of maximal dimension orbits of the Borel subgroup B subset ofG(C) in the complex algebraic variety G(C)/K-C. Moreover, if delta (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. [References: 17]
展开▼
