  The Verma modules over the quantum groups $\mathrm U_q(\mathfrak{gl}_{l +
1})$ for arbitrary values of $l$ are analysed. The explicit expressions for the
action of the generators on the elements of the natural basis are obtained. The
corresponding representations of the quantum loop algebras $\mathrm
U_q(\mathcal L(\mathfrak{sl}_{l + 1}))$ are constructed via Jimbo's
homomorphism. This allows us to find certain representations of the positive
Borel subalgebras of $\mathrm U_q(\mathcal L(\mathfrak{sl}_{l + 1}))$ as
degenerations of the shifted representations. The latter are the
representations used in the construction of the so-called $Q$-operators in the
theory of quantum integrable systems. The interpretation of the corresponding
simple quotient modules in terms of representations of the $q$-deformed
oscillator algebra is given.
