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We discuss some natural maps from a unitary group U(n) to a smaller groupU(n-m) (these maps are versions of the Livshic characteristic function).  We calculate explicitly the direct images of the Haar measure under somemaps.  We evaluate some matrix integrals over classical groups and some symmetricspaces (values of the integrals are products of Gamma-functions). Theseintegrals generalize Hua Loo Keng integrals. We construct inverse limits ofunitary groups equipped with analogues of the Haar measure and evaluate someintegrals over these inverse limits.Comment: 20 pages; the final version; evaluation of some normalizing constants  is adde

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