Integrable G-invariant Hamiltonian systems and homogeneous spaces with simple spectrum (Q1104614)

Let G be a connected semisimple real Lie group and K its compact subgroup. In the paper under review the following theorem (with scheme of a proof) is given. Theorem. The following properties are equivalent: (1) All G-invariant Hamilton systems on T*(G/K) are completely integrable in the Noether integrals (i.e. integrals obtained from functions on \(g*\) through momentum map). (2) The spectrum of the unitary representation of G by left translations in \(L^ 2(G/K)\) is simple. For compact groups G this result was stated earlier in the papers [\textit{V. Guillemin} and \textit{S. Sternberg}, J. Differ. Geom. 19, 31-56 (1984; Zbl 0548.58017); \textit{I. V. Mikityuk}, Mat. Sb., Nov. Ser. 129(171), No.4, 514-534 (1986; Zbl 0621.70005)].