  The degenerate Lie group is a semidirect product of the Borel subgroup with
the normal abelian unipotent subgroup. We introduce a class of the highest
weight representations of the degenerate group of type A, generalizing the
PBW-graded representations of the classical group. Following the classical
construction of the flag varieties, we consider the closures of the orbits of
the abelian unipotent subgroup in the projectivizations of the representations.
We show that the degenerate flag varieties $\Fl^a_n$ and their
desingularizations $R_n$ can be obtained via this construction. We prove that
the coordinate ring of $R_n$ is isomorphic to the direct sum of duals of the
highest weight representations of the degenerate group. In the end, we state
several conjectures on the structure of the highest weight representations.
