It is well known $\newcommand{\Diff}{\operatorname{Diff}}$ that the group $\Diff(S^1)$ of smooth diffeomorphisms of the circle behaves in many ways like $SL(2,\mathbb R)$. For example, if $S^1\subseteq \Diff(S^1)$ is the group of rotations then $\Diff(S^1)/S^1$ is an infinite dimensional complex domain (apparently this is due to Kirillov and Yuriev). This is analogous to the symmetric space $SL(2,\mathbb R)/S^1$ which is the complex upper half plane.
Now $SL(2,\mathbb R)$ has some very useful decompositions: the Iwasawa decomposition (or the $KAN$ decomposition) and the $KAK$ decomposition.
Does $\Diff(S^1)$ also have $KAN$ and $KAK$ decompositions?