Hi. Is there a Haar measure or equivalent on infinite dimensional Lie groups? I've been playing around with $Diff(S^1)$, and at least a direct approach seems quite hopeless. It goes something like this:


Def. element on the group by "Euler coordinates",


$g \doteq \prod\limits\_{i=-\infty}^{\infty} e^{\omega^i X\_i}$, with $\left[ X\_i ,X\_j \right] = (j-i)X\_{i+j}$.


Now I could define a (left invariant) Maurer-Cartan form as $\Omega\_L \doteq g^{-1} dg = X\_i \otimes \theta^i$, where $\theta^i = \mathcal L^i\_j d\omega^j$. Then the Haar measure is


$d\mu (g) \doteq ||\mathcal L || \bigwedge\limits\_i d\omega^i$.


Elements of $\mathcal L$ can be written as


$\mathcal L^i\_j = \left( \prod\limits\_{n=\infty}^{j+1} \exp(-\omega^n adX\_n) \right)^i\_j$


Clearly the determinant $||\mathcal L ||$ will be horrible... is there any hope for a manageable explicit expression? I couldn't find any literature on the subject (yet), so I'd appreciate any hints to the right direction.


EDIT: umm and of course the whole question of existence of such a measure should probably addressed...


EDIT 2: I realized that the question setup is a bit misleading: I'm actually looking for a measure on the *Virasoro group* (with zero central charge), i.e. the Lie group corresponding to the algebra above... maybe the Shavgulidze measure has something to do with it, I don't know...



Haar measure on infinite dimensional Lie groups?
There is something called the Shavgulidze, or the Malliavin-Shavgulidze measure on Diff of smooth manifolds. You can find a discussion in Differentiable measures and the Malliavin calculus (p. 397, available on google books). It is not quite invariant, but quasi-invariant.


