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Article

G<sub>a</sub><sup>M</sup> degeneration of flag varieties

Selecta Mathematica, New Series. 2012. Vol. 18. No. 3. P. 513-537.

Let Fλ be a generalized flag variety of a simple Lie group G embedded into the projectivization of an irreducible G-module Vλ. We define a flat degeneration Fλa, which is a GaM variety. Moreover, there exists a larger group Ga acting on Fλa, which is a degeneration of the group G. The group Ga contains GaM 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 to the product of projective spaces. The defining ideal of Fλa is generated by the set  of degenerate Plüker relations. We prove that the coordinate ring of Fλa is isomorphic to a direct sum of dual PBW-graded g-modules. We also prove that there exist bases in multi-homogeneous components of the coordinate rings, parametrized by the semistandard PBW-tableux, which are analogues of semistandard tableaux.