This question is a refined version of http://mathoverflow.net/questions/97659/representations-of-infinite-dimensional-lie-algebras-as-vector-fields-on-manifold
I'm interested in the finite dimensional homogeneous spaces of $Diff(S^1)$. There are several papers concerned with infinite dimensional homogeneous spaces such as $Diff(S^1)/S^1$, but I can't find anything of the finite dimensional spaces, except the paper by Cartan from 1905, as hinted by Robert Bryant in the above link... From R.B's reply in the link:
"You could try É. Cartan's papers on infinite pseudogroups (mostly appearing 1904-05). In particular, see Paragraph 57 of Sur la structure des groupes infinis de transformation (suite). There, for example, he proves that the (pseudo-)group of diffeomorphisms of the line has three distinct (i.e., nonequivalent) 2-dimensional homogeneous spaces and seven distinct 3-dimensional homogeneous spaces, etc."
Basically I'm interested in the two dimensional spaces... it's just that I don't know French, and the exposition in Cartan's paper seems quite brief anyway. Surely there must be more modern works about the subject?
Also, it's not very clear to me how to construct these homogeneous spaces as cosets...
EDIT: Here's an example from Cartan's 1905 paper (I still don't know French but I was able to decipher that much):
Suppose we have vector fields as a Lie algebra of $Diff(\mathbb R)$ (Cartan is doing it on the line, but the circle is similar),
$l_n = x^{n+1} \partial_x$.
Cartan obtained corresponding vector fields on three dimensional homogeneous spaces of $Diff(\mathbb R)$. On one of the seven homogeneous spaces they are
$l_n = x^{n+1}\partial _x+\frac{y}{(n+1)x^n}\partial x-(n+1) x^n y\partial _y+\left(\frac{z}{(n+1)}x^{-n}-\frac{n}{n+1}x^{-n-1}\right)\partial y-(n+1)(n+x z)x^{n-1}\partial _z$,
but what is the homogeneous space?space (by which I mean, can it be expressed as a quotient)?!
Here's a link to to the relevant pages in Cartan's paper: http://goo.gl/bJXfm
EDIT: fixed some typos Fixed a misunderstanding regarding Cartan's notation in the formulaeformula (see RB's answer below). I also expanded the vectors vector fields in a basis by Taylor expanding the function used by Cartan, $f(x) = \sum c_n x^{n+1}$. That way it's easy to see that they both satisfy the Witt algebra, $[l_n, l_m] = -(n-m)l_{n+m}$...
