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

Abstract

Let $\Fl_\lambda$ be a generalized flag variety of a simple Lie group $G$ embedded into the projectivization of an irreducible $G$-module $V_\lambda$. We define a flat degeneration $\Fl_\lambda^a$, which is a ${\mathbb G}^M_a$ variety. Moreover, there exists a larger group $G^a$ acting on $\Fl_\lambda^a$, 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 embedded into the product of Grassmanians and thus to the product of projective spaces. The defining ideal of $\Fl^a_\lambda$ is generated by the set of degenerate Pl\" ucker relations. We prove that the coordinate ring of $\Fl_\lambda^a$ 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 tableux.