CONDITIONAL HAAR MEASURES ON CLASSICAL COMPACT GROUPS

We give a probabilistic proof of the Weyl integration formula on U(n), the unitary group with dimension it. This relies on a suitable definition of Haar measures conditioned to the existence of a stable subspace with any given dimension p. The developed method leads to the following result: for this conditional measure, writing Z(U)((p)) for the first nonzero derivative of the characteristic polynomial at 1, Z(U)((p))/p! =(law) Pi(n-p)(l=1)(1-X(l)), the X(l)'s being explicit 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. Similar limit theorems are given for the orthogonal and symplectic groups, relying on results of Killip and Nenciu.