Loops in Noncompact Groups of Hermitian Symmetric Type and Factorization

Caine A,Pickrell D

doi:10.4172/1736-4337.1000233

Abstract

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 showed that for a constant loop there is a unique Birkhoff factorization if and only if 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.

Highlights

Finite dimensional Riemannian symmetric spaces come in dual pairs, one of compact type and one of noncompact type
K where U is the universal covering of the identity component of the isometry group of the compact type symmetric space X U / K, G is the complexification of U, and G 0 is a covering of the isometry group for the dual noncompact symmetric space X 0 = G 0 / K
The main purpose of this paper is to investigate Birkhoff factorization and “root subgroup factorization" for the loop group of G 0, assuming G 0 is of Hermitian symmetric type so that X0 and X are Hermitian symmetric spaces

Summary

Introduction

Finite dimensional Riemannian symmetric spaces come in dual pairs, one of compact type and one of noncompact type. One might still naively expect that there could be a relatively transparent way to parameterize the intersections of the Birkhoff components with LG 0 (as in the finite dimensional case, and in the case of loops into compact groups, e.g., using root subgroup factorization).

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Results

Conclusion