TITLE: Iwasawa and $KAK$ Decomposition for Diff$(S^1)$
QUESTION [10 upvotes]: 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?

REPLY [7 votes]: At least for the $KAN$ decomposition, there is a candidate.
Namely take $K$ the rotation group, $A$ the group of hyperbolic elements of $PSL(2,\mathbb{R})$ fixing $0$ and $\infty$, and $N$ the group of diffeos fixing $0$ and tangent to the identity there.
Then $G=KAN$ is the group $Diff_+(\mathbb{S}^1)$, and the decomposition of elements is unique, as is easily checked.
No idea yet for the $KAK$ decomposition.