Conditional Haar measures on classical compact groups

Abstract

We give a probabilistic proof of the Weyl integration formula on U(n), the unitary group with dimension n. 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, $$\frac{Z_{U}^{(p)}}{p!}\stackrel{\mathrm{law}}{=}\prod_{\ell =1}^{n-p}(1-X_{\ell})$$ 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.