Noncompact Groups of Hermitian Symmetric Type and Factorization

Abstract

We investigate Birkhoff (or triangular) factorization and (what we propose to call) root subgroup factorization for elements of a noncompact simple Lie group G 0 of inner type. For compact groups root subgroup factorization is related to Bott-Samelson desingularization, and many striking applications have been discovered by Lu ([3]). In this paper, in the inner noncompact case, we obtain parallel characterizations of the Birkhoff components of G 0 and an analogous construction of root subgroup coordinates for the Birkhoff components. As in the compact case, we show that the restriction of Haar measure to the top Birkhoff component is a product measure in root subgroup coordinates.

Complex groups and root subgroup factorization

“…There is a substantial literature on (a number of slightly different versions of) root subgroup factorization for a unitary form U of G. Some of the main references are Kac-Peterson [7] (following seminal ideas of Steinberg on the structure of finite groups of Lie type), Bott-Samelson [3] (on the desingularization of Schubert varieties), Soibelman [13] and Lu [10] (Poisson geometry). For real noncompact groups, see [4]. One motivation for returning to this topic in a complex setting is the existence of a generalization to loop groups which has nontrivial consequences for the calculation of Toeplitz determinants, see [2].…”