Degenerate Group of Type $A$: Representations and Flag Varieties

We consider the degeneration of a simple Lie group which is a semidirect product of its Borel subgroup and a normal Abelian unipotent subgroup. We introduce a class of highest weight representations of the degenerate group of type $A$, generalizing the construction of PBW-graded representations of the classical group (PBW is an abbreviation for “Poincaré–Birkhoff–Witt”). Following the classical construction of flag varieties, we consider the closures of orbits of the Abelian unipotent subgroup in projectivizations of the representations. We show that the degenerate flag varieties $\mathscr{F}^a_n$ and their desingularizations $R_n$ can be obtained via this construction. We prove that the coordinate ring of $R_n$ is isomorphic as a vector space to the direct sum of the duals of the highest weight representations of the degenerate group. At the end we state several conjectures on the structure of the highest weight representations of the degenerate group of type $A$.