Loops in Noncompact Groups of Hermitian Symmetric Type and Factorization

In previous work with Pittmann-Polletta, 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 type semisimple Lie group of Hermitian symmetric type. In previous work we showedthat for a constant loop there is a unique Birkhoff factorization if and onlyif there is a root subgroup factorization. However for loops, 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.

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