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\({\mathbb{G}_a^M}\) degeneration of flag varieties. (English) Zbl 1267.14064
Let \(G\) be a simple Lie group. The flag varieties associated with \(G\) are the homogeneous spaces \(G/P\) by the parabolic subgroups. These varieties can be realized as \(G\)-orbits \(G[v_{\lambda}]\subseteq\mathbb{P}(V_{\lambda})\) in the projectivization of the simple \(G\)-modules \(V_{\lambda}\) with the highest weight vector \(v_{\lambda}\). Let us denote \(\mathcal{F}_{\lambda}=G[v_{\lambda}]\).
In this paper, the author introduces a family of varieties \(\mathcal{F}_{\lambda}^a\), which are flat degenerations of \(\mathcal{F}_{\lambda}\). These varieties are equipped with an action of the \(M\)-fold product \(\mathbb{G}_a^M\) of the additive group of the ground field with an open orbit, where \(M\) is the dimension of a maximal unipotent subgroup of \(G\). Moreover, there exists a larger group \(G^a\) acting on \(\mathcal{F}_{\lambda}^a\), which is a degeneration of the group \(G\) and which contains \(\mathbb{G}_a^M\) as a normal subgroup.
If \(G\) is of type \(A\), then the degenerate flag varieties can be embedded into the product of Grassmannians and thus into the product of projective spaces. The defining ideal of \(\mathcal{F}_{\lambda}^a\) is generated by the set of degenerate Plücker relations. It is proved that the coordinate ring of \(\mathcal{F}_{\lambda}^a\) is isomorphic to a direct sum of dual PBW-graded modules. Also it is shown that there exists bases in multi-dimensional components of the coordinate rings parametrized by the semistandard PBW-tableaux.

MSC:
14L35 Classical groups (algebro-geometric aspects)
17B45 Lie algebras of linear algebraic groups
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[1] Arzhantsev, I.: Flag varieties as equivariant compactifications of $${\(\backslash\)mathbb{G}_a\^n}$$ , arXiv:1003.2358 · Zbl 1217.14032
[2] Arzhantsev, I., Sharoiko, E.: Hassett-Tschinkel correspondence: modality and projective hypersurfaces, arXiv:0912.1474 · Zbl 1248.14053
[3] Caldero P.: Toric degenerations of Schubert varieties. Transform. Groups 7(1), 51–60 (2002) · Zbl 1050.14040 · doi:10.1007/BF01253464
[4] Feigin E.: The PBW filtration. Represent. Theory 13, 165–181 (2009) · Zbl 1229.17026 · doi:10.1090/S1088-4165-09-00349-5
[5] Feigin, E.: The PBW filtration, Demazure modules and toroidal current algebras. SIGMA 4, 070, 21 (2008) · Zbl 1215.17015
[6] Fulton W.: Young Tableaux, with Applications to Representation Theory and Geometry. Cambridge University Press, Cambridge (1997) · Zbl 0878.14034
[7] Feigin, B., Feigin, E., Littelmann, P.: Zhu’s algebras, C 2-algebras and abelian radicals, arXiv:0907.3962 · Zbl 1292.17024
[8] Feigin, E., Fourier, G., Littelmann, P.: PBW filtration and bases for irreducible modules in type A n , arXiv:1002.0674 · Zbl 1237.17011
[9] Hartshorne R.: Algebraic Geometry, GTM, No. 52. Springer, New York (1977) · Zbl 0367.14001
[10] Feigin E., Littelmann P.: Zhu’s algebra and the C 2-algebra in the symplectic and the orthogonal cases. J. Phys. A Math. Theor. 43, 135206 (2010) · Zbl 1214.81243 · doi:10.1088/1751-8113/43/13/135206
[11] Gonciulea N., Lakshmibai V.: Degenerations of flag and Schubert varieties to toric varieties. Transform. Groups 1(3), 215–248 (1996) · Zbl 0909.14028 · doi:10.1007/BF02549207
[12] Hassett B., Tschinkel Y.: Geometry of equivariant compactifications of $${\(\backslash\)mathbb{G}\^n_a}$$ . Int. Math. Res. Notices 20, 1211–1230 (1999) · Zbl 0966.14033 · doi:10.1155/S1073792899000665
[13] Kumar S.: Kac-Moody Groups, Their Flag Varieties and Representation Theory, Progress in Mathematics, vol. 204. Birkhäuser, Boston (2002) · Zbl 1026.17030
[14] Kumar S.: The nil Hecke ring and singularity of Schubert varieties. Invent. Math. 123, 471–506 (1996) · Zbl 0863.14031
[15] Lakshmibai V.: Degenerations of flag varieties to toric varieties. C. R. Acad. Sci. Paris 321, 1229–1234 (1995) · Zbl 0858.14026
[16] Vinberg, E.: On some canonical bases of representation spaces of simple Lie algebras, conference talk, Bielefeld (2005)
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