  The structure and representations of the quantum general linear supergroup
GLq(m|n) are studied systematically by investigating the Hopf superalgebra Gq
of its representative functions. Gq is factorized into $Gq^{\pi} Gq^{\bar\pi}$,
and a Peter - Weyl basis is constructed for each factor. Parabolic induction
for the quantum supergroup is developed. The underlying geometry of induced
representations is discussed, and an analog of Frobenius reciprocity is
obtained. A quantum Borel - Weil theorem is proven for the covariant and
contravariant tensorial irreps, and explicit realizations are given for classes
of tensorial irreps in terms of sections of quantum super vector bundles over
quantum projective superspaces.
