Degenerate $SL_n$: representations and flag varieties

TL;DR

This paper introduces a new class of highest weight representations for the degenerate Lie group of type A, constructs associated flag varieties, and explores their algebraic and geometric properties, including desingularizations and coordinate rings.

Contribution

It generalizes PBW-graded representations, constructs degenerate flag varieties and their desingularizations, and establishes isomorphisms with coordinate rings, advancing understanding of degenerate Lie groups.

Findings

Degenerate flag varieties can be obtained via orbit closures.

Coordinate ring of desingularizations is isomorphic to duals of highest weight representations.

Several conjectures on the structure of these representations are proposed.

Abstract

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.