Loops in noncompact groups and factorization

In [11] we showed that a loop in a simply connected compact Lie group $\dot{U}$ 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 type semisimple Lie group ${\dot{G}}_0$ of inner type. In [4] we showed that for an element of ${\dot{G}}_0$, 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 ${\dot{G}}_0$, while a root subgroup factorization implies a unique Birkhoff factorization, there are several obstacles to the converse. As in the compact case, root subgroup factorization is intimately related to factorization of Toeplitz determinants.