Compact Symmetric Spaces, Triangular Factorization, and Poisson Geometry

LetX be a simply connected compact Riemannian symmetric space, let U be the universal covering group of the identity component of the isometry group of X , and let g denote the complexification of the Lie algebra of U , g = u. Each u-compatible triangular decomposition g = n − + h + n+ determines a Poisson Lie group structure πU on U . The Evens-Lu construction ([EL]) produces a (U, πU )-homogeneous Poisson structure on X . By choosing the basepoint in X appropriately, X is presented as U/K where K is the fixed point set of an involution which stabilizes the triangular decomposition of g. With this presentation, a connection is established between the symplectic foliation of the Evens-Lu Poisson structure and the Birkhoff decomposition of U/K. This is done through reinterpretation of results of Pickrell ([Pi]). Each symplectic leaf admits a natural torus action. It is shown that the action is Hamiltonian and the momentum map is computed using triangular factorization. Finally, local formulas for the Evens-Lu Poisson structure are displayed in several examples.