  In this paper we construct explicitly natural (from the geometrical point of
view) Fock space representations (contragradient Verma modules) of the
quantized enveloping algebras. In order to do so, we start from the Gauss
decomposition of the quantum group and introduce the differential operators on
the corresponding $q$-deformed flag manifold (asuumed as a left comodule for
the quantum group) by a projection to it of the right action of the quantized
enveloping algebra on the quantum group. Finally, we express the
representatives of the elements of the quantized enveloping algebra
corresponding to the left-invariant vector fields on the quantum group as
first-order differential operators on the $q$-deformed flag manifold.
