The classical coarea formula provides us a possibility to reduce the integration over some set to the integration over the slices of this set. For example, we can reduce the integration over a unit ball to the integration over spheres: $$ \int\limits_{B(0,1)} f(x) \, dx = \int\limits_{0}^{1} \int\limits_{S(0,t)} f(x)\,dS \, dt. $$ I'm looking for the generalization of this result to the case of topological groups. For example, is there some similar formula that allows us to reduce the integral over $GL(n)$ to an integral over $O(n)$ and etc?
I'm also looking for such decomposition of integral over the additive group of symmetric matrices. I want to reduce the integration to the integration over the set of eigenvalues.