Hua-type integrals over unitary groups and over projective limits of unitary groups

Abstract

We discuss some natural maps from a unitary group ${\rm U}(n)$ to a smaller group ${\rm U}(n-m)$. (These maps are versions of the Livšic characteristic function.) We calculate explicitly the direct images of the Haar measure under some maps. We evaluate some matrix integrals over classical groups and some symmetric spaces. (Values of the integrals are products of $\Gamma$-functions.) These integrals generalize Hua integrals. We construct inverse limits of unitary groups equipped with analogues of Haar measure and evaluate some integrals over these inverse limits.