
Journal of Lie Theory 28 (2018), No. 4, 10951118 Copyright Heldermann Verlag 2018 Complex Groups and Root Subgroup Factorization Doug Pickrell Mathematics Department, University of Arizona, Tucson, AZ 85721, U.S.A. pickrell@math.arizona.edu Root subgroup factorization is a refinement of triangular (or LDU) factorization. For a complex reductive Lie group, and a choice of reduced factorization of the longest Weyl group element, the forward map from root subgroup coordinates to triangular coordinates is polynomial. We show that the forward map is injective on its set of regular points and that the inverse is rational. There is an algorithm for the inverse (involving LDU factorization), and a related explicit formula for Haar measure in root subgroup coordinates. In classical cases there are preferred reduced factorizations of the longest Weyl group elements, and conjecturally in these cases there are closed form expressions for root subgroup coordinates. Keywords: Complex reductive group, triangular factorization, root subgroup factorization MSC: 22E67 [ Fulltextpdf (177 KB)] for subscribers only. 