Well I left out some of it, hoping that he'd be able to connect the dots. *dripping sarcasm*
What I meant to imply was a probabilistic proof of the Weyl integration formula on the unitary group. This relies on a suitable definition of Haar measures conditioned to the existence of a stable subspace with any given dimension. The developed method leads to the following: for this conditional measure, writing $Z_U^{(p)}$ for the first non-zero derivative of the characteristic polynomial at 1, $Z_U^{(p)}$ can be decomposed as an explicit product of $n-p$ independent random variables. This implies a central limit theorem for $\log Z_U^{(p)}$ and asymptotics for the density of $Z_U^{(p)}$ near 0. I mean, duh. Similar limit theorems apply for the orthogonal and symplectic groups. Now if you are mapping x->Ux, Let {ei, e2s ... en} be the complete orthonormal system of Rn, so that if xn = <x, en >=