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Article

Degenerate group of type A: Representations and flag varieties

Functional Analysis and Its Applications. 2014. Vol. 48. No. 1. P. 59-71.

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.