Loops in Noncompact Groups of Hermitian Symmetric Type and Factorization

Caine A and Pickrell D

Abstract

In studies of Pittmann, we showed that a loop in a simply connected compact Lie group has a unique Birkhoff (or triangular) factorization if and only if the loop has a unique root subgroup factorization (relative to a choice of a reduced sequence of simple reflections in the affine Weyl group). In this paper our main purpose is to investigate Birkhoff and root subgroup factorization for loops in a noncompact semisimple Lie group of Hermitian symmetric type. In literature of caine, we showed that for an element of, i.e. a constant loop, there is a unique Birkhoff factorization if and only if there is a root subgroup factorization. However for loops in, while a root subgroup factorization implies a unique Birkhoff factorization, the converse is false. As in the compact case, root subgroup factorization is intimately related to factorization of Toeplitz determinants.