Compact symmetric spaces, triangular factorization, and Cayley coordinates

TL;DR

This paper derives explicit formulas for triangular factorizations of elements in compact symmetric spaces using Cayley coordinates, linking geometric, algebraic, and Poisson structures.

Contribution

It provides an explicit Cayley coordinate formula for the diagonal part of the triangular factorization in symmetric spaces, enhancing understanding of their geometric and algebraic properties.

Findings

Explicit formula for d(g) in Cayley coordinates.

Identification of connected components in the symmetric space.

Calculation of a moment map for a torus action.

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.