Poisson Geometry and Matrix Factorization

Riemannian symmetric spaces admit representations as coset spaces of Lie groups and, in some cases, admit totally geodesic embeddings into those groups. Algebraic decompositions of the groups induce decompositions of the symmetric spaces. In this lecture we discuss homogeneous Poisson structures on symmetric spaces which are related to the Iwasawa, Bruhat, and and Birkhoff decompsositions. In the classical case of the complex grassmannian, the Lie group of interest is the special unitary group SU(n) and the relevant decompositions are of its complexification, SL(n,C). The Iwasawa decomposition is that given by the factorization resulting from the Gram-Schmidt process and the Birkhoff decomposition is that arising from the matrix factorization resulting Gaussian elimination. We describe the geometry of the symplectic foliation of these Poisson structures and exhibit momentum maps for natural torus actions on the symplectic leaves. Through Duistermaat-Heckman techniques these momentum maps have found application to the computation of integral formulas for the diagonal distribution of the invariant measure of a compact symmetric space. This is joint work with Doug Pickrell.