Compact symmetric spaces , triangular factorization , and Cayley coordinates Derek Habermas

Let U/K represent a connected, compact symmetric space, where θ is an involution of U that fixes K, φ : U/K → U is the geodesic Cartan embedding, and G is the complexification of U. We investigate the intersection, studied by Pickrell, of φ(U/K) with the Birkhoff decomposition of G corresponding to a θ-stable triangular, or LDU, factoriztion of Lie(G). When g ∈ φ(U/K) is generic, the factorization g = ldu is unique. In this paper we present explicit formulas for d in Cayley coordinates for several classes of symmetric spaces, and use them to compute the connected components of the " top level, " or generic part, of φ(U/K). These formulas are also useful in explicitly calculating Evens-Lu Poisson structures on U/K, also studied by Foth, Otto, and Caine, and for computing related integrals using elementary techniques.