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
Abstract

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.

Keywords
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
Milestones
Received: 29 September 2008
Revised: 11 September 2011
Accepted: 14 September 2011
Published: 28 November 2011
Authors
Derek Habermas
Department of Mathematics
SUNY Potsdam
44 Pierrepont Avenue
Potsdam, NY 13676
United States