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.
Similar content being viewed by others
References
Arzhantsev, I.: Flag varieties as equivariant compactifications of \({\mathbb{G}_a^n}\) , arXiv:1003.2358
Arzhantsev, I., Sharoiko, E.: Hassett-Tschinkel correspondence: modality and projective hypersurfaces, arXiv:0912.1474
Caldero P.: Toric degenerations of Schubert varieties. Transform. Groups 7(1), 51–60 (2002)
Feigin E.: The PBW filtration. Represent. Theory 13, 165–181 (2009)
Feigin, E.: The PBW filtration, Demazure modules and toroidal current algebras. SIGMA 4, 070, 21 (2008)
Fulton W.: Young Tableaux, with Applications to Representation Theory and Geometry. Cambridge University Press, Cambridge (1997)
Feigin, B., Feigin, E., Littelmann, P.: Zhu’s algebras, C 2-algebras and abelian radicals, arXiv:0907.3962
Feigin, E., Fourier, G., Littelmann, P.: PBW filtration and bases for irreducible modules in type A n , arXiv:1002.0674
Hartshorne R.: Algebraic Geometry, GTM, No. 52. Springer, New York (1977)
Feigin E., Littelmann P.: Zhu’s algebra and the C 2-algebra in the symplectic and the orthogonal cases. J. Phys. A Math. Theor. 43, 135206 (2010)
Gonciulea N., Lakshmibai V.: Degenerations of flag and Schubert varieties to toric varieties. Transform. Groups 1(3), 215–248 (1996)
Hassett B., Tschinkel Y.: Geometry of equivariant compactifications of \({\mathbb{G}^n_a}\) . Int. Math. Res. Notices 20, 1211–1230 (1999)
Kumar S.: Kac-Moody Groups, Their Flag Varieties and Representation Theory, Progress in Mathematics, vol. 204. Birkhäuser, Boston (2002)
Kumar S.: The nil Hecke ring and singularity of Schubert varieties. Invent. Math. 123, 471–506 (1996)
Lakshmibai V.: Degenerations of flag varieties to toric varieties. C. R. Acad. Sci. Paris 321, 1229–1234 (1995)
Vinberg, E.: On some canonical bases of representation spaces of simple Lie algebras, conference talk, Bielefeld (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Feigin, E. \({\mathbb{G}_a^ M}\) degeneration of flag varieties. Sel. Math. New Ser. 18, 513–537 (2012). https://doi.org/10.1007/s00029-011-0084-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00029-011-0084-9