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Selecta Mathematica

, Volume 18, Issue 3, pp 513–537 | Cite as

\({\mathbb{G}_a^ M}\) degeneration of flag varieties

  • Evgeny FeiginEmail author
Article

Abstract

Let \({\mathcal{F}_\lambda}\) be a generalized flag variety of a simple Lie group G embedded into the projectivization of an irreducible G-module V λ . We define a flat degeneration \({\mathcal{F}_\lambda^a}\) , which is a \({\mathbb{G}^M_a}\) variety. Moreover, there exists a larger group G a acting on \({\mathcal{F}_\lambda^a}\) , which is a degeneration of the group G. The group G a contains \({\mathbb{G}^M_a}\) as a normal subgroup. If G is of type A, then the degenerate flag varieties can be embedde‘d into the product of Grassmannians and thus to the product of projective spaces. The defining ideal of \({\mathcal{F}_\lambda}\) is generated by the set of degenerate Plücker relations. We prove that the coordinate ring of \({\mathcal{F}_\lambda^a}\) is isomorphic to a direct sum of dual PBW-graded \({\mathfrak{g}}\) -modules. We also prove that there exists bases in multi-homogeneous components of the coordinate rings, parametrized by the semistandard PBW-tableux, which are analogs of semistandard tableaux.

Keywords

Lie groups Flag varieties Degeneration Plücker relations 

Mathematics Subject Classification (2000)

14L35 17B45 

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Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Department of MathematicsNational Research University Higher School of EconomicsMoscowRussia
  2. 2.Tamm Theory DivisionLebedev Physics InstituteMoscowRussia

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