# Relation between volume of reduced space and phase space

Let $G$ ba a compat Lie group and $\frak g$ be its Lie algebra, then by Marsden-Weinstein reduction theory we know that if $J:M\to \frak g^*$ be its equivariant moment map then the reduced space is $$S=J^{-1}(\mu)/G_\mu$$ where $\mu\in \frak g^*$ and $G_\mu$ is isotropy group at point $\mu$ .

What is the relation between $Vol(S)$ and $Vol(M)$

Not a lot. Instead you should generalize $Vol(M)$, the pushforward of Liouville measure to a point, to $Vol_G(M)$, the pushforward along the moment map. I'll assume $G=T$ for convenience. The result is piecewise-polynomial $f$ times Lebesgue measure on the image, and $Vol(S) = f(\mu)$. This is much of the content of the original Duistermaat-Heckman paper, which you should read before asking further questions in this direction.