Abstract
Let G=K be a semisimple orbit of the adjoint representation of a real connected reductive Lie group G. Let K1 be any closed subgroup of K containing the commutant of the identity component of K. We prove that the geodesic flow on the symplectic manifold T ⁄ (G=K1), corresponding to a G-invariant pseudo-Riemannian metric on G=K1 which is in- duced by a bi-invariant pseudo-Riemannian metric on G, is completely integrable in the class of real analytic functions, polynomial in momenta. To this end we study the Poisson geometry of the space of G-invariant functions on T ⁄ (G=K) using a one-parameter family of moment maps.
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