A note on Iwasawa-type decomposition

We study the Iwasawa-type decomposition of an open subset of SL(n,C) as SU(p,q)AN. We show that the dressing action of SU(p,q) is globally defined on the space of admissible elements in AN. We also show that the space of admissible elements is a multiplicative subset of AN. We establish a geometric criterion: the symmetrization of an admissible element maps the positive cone in C^n into itself.