G _a ^M degeneration of flag varieties

Let F _λ 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 F _λ ^a variety. Moreover, there exists a larger group G ^a acting on F _λ ^a, which is a degeneration of the group G. The group G ^a contains G _a ^M 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 F _λ ^a is generated by the set of degenerate Plücker relations. We prove that the coordinate ring of F _λ ^a is isomorphic to a direct sum of dual PBW-graded 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.