Dressing Orbits of Harmonic Maps

this paper is to describe some interactions between these two approaches. Our starting point is the fact that underlying all of the above results is the existence of Iwasawa type decompositions of the loop groups and algebras concerned. On the one hand, the Lax equations mentioned above arise from an Iwasawa decomposition of certain twisted loop algebras via the Adler--Kostant--Symes scheme [5]. On the other hand, the loop group action of Uhlenbeck is essentially the dressing action arising from the Iwasawa decompositions of the corresponding loop groups [9]. Moreover, a bridge beween these constructions is provided by Symes's formula for the solution of the Lax equations: in this construction, first applied by Symes [21] to solve the ope