A Note on Iwasawa-Type Decomposition

In Poisson geometry, the groups SU p, q andAN the upper-triangular subgroup of SL n, with real positive diagonal entries are naturally dual to each other 1 . Therefore, it is important to know the geometry of the orbits of the dressing action. We show that the right dressing action of SU p, q is globally defined on the open subset of the so-called admissible elements of AN see Section 2 . We also show that the admissible elements can be characterized as follows: these are exactly those elements of AN whose symmetrization maps the closure of the positive cone in n into the positive cone. In addition, we establish a useful fact that the set of admissible elements is a multiplicative subset of AN.