Compact symmetric spaces, triangular factorization, and Cayley coordinates
Abstract
Let U/K represent a connected, compact symmetric space, where theta is an involution of U that fixes K, phi: U/K to U is the geodesic Cartan embedding, and G is the complexification of U. We investigate the intersection of phi(U/K) with the Bruhat decomposition of G corresponding to a theta-stable triangular, or LDU, factorization of the Lie algebra of G. When g in phi(U/K) is generic, the corresponding factorization g=ld(g)u is unique, where l in N^-, d(g) in H, and u in N^+. In this paper 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 phi(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.
Cite
@article{arxiv.0808.2251,
title = {Compact symmetric spaces, triangular factorization, and Cayley coordinates},
author = {Derek Habermas},
journal= {arXiv preprint arXiv:0808.2251},
year = {2011}
}
Comments
19 pages: Main proof entirely rewritten, sections reorganized, exposition made more precise and concise. To appear in Pacific Journal of Mathematics