  We discuss some natural maps from a unitary group U(n) to a smaller group
U(n-m) (these maps are versions of the Livshic 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 Loo Keng integrals. We construct inverse limits of
unitary groups equipped with analogues of the Haar measure and evaluate some
integrals over these inverse limits.
