Degenerate group of type A: Representations and flag varieties

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2 Citations (Scopus)

Abstract

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 Fna and their desingularizations Rn can be obtained via this construction. We prove that the coordinate ring of Rn 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.

Original languageEnglish
Pages (from-to)59-71
Number of pages13
JournalFunctional Analysis and its Applications
Volume48
Issue number1
DOIs
Publication statusPublished - 2014
Externally publishedYes

Keywords

  • flag varieties
  • highest weight modules
  • Lie algebras
  • PBW filtration

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