Vol. 253, No. 1, 2011

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Compact symmetric spaces, triangular factorization, and Cayley coordinates

Derek Habermas

Vol. 253 (2011), No. 1, 57–73

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 of ϕ(U∕K) with the Bruhat decomposition of G corresponding to a 𝜃-stable triangular, or LDU, factorization of the Lie algebra of G. When g ϕ(U∕K) is generic, the corresponding factorization g = ld(g)u is unique, where l N , d(g) H, and u N+. We present an explicit formula for d in Cayley coordinates, compute it in several types of symmetric spaces, and use it to identify representatives of the connected components of the generic part of ϕ(U∕K). This formula calculates a moment map for a torus action on the highest dimensional symplectic leaves of the Evens–Lu Poisson structure on U∕K.

compact symmetric space, triangular factorization, ldu factorization, Bruhat decomposition, Cayley map, Cayley coordinates, symplectic leaves, compute, computation, concrete, classical, connected component, Cartan embedding, antidiagonal, antitranspose
Mathematical Subject Classification 2000
Primary: 53C35
Secondary: 43A85
Received: 29 September 2008
Revised: 11 September 2011
Accepted: 14 September 2011
Published: 28 November 2011
Derek Habermas
Department of Mathematics
SUNY Potsdam
44 Pierrepont Avenue
Potsdam, NY 13676
United States