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

Let ${\text{Fl}}_{\lambda }$ 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 ${\text{Fl}}_{\lambda }^a$, which is a ${\mathbb{G}}_a^M$ variety. Moreover, there exists a larger group $G^a$ acting on ${\text{Fl}}_{\lambda }^a$, which is a degeneration of the group $G$. The group $G^a$ contains ${\mathbb G}^M_a$asanormalsubgroup\mathrm{.}If$G$isoftype$A$, 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 ${\text{Fl}}_{\lambda }^a$ is generated by the set of degenerate Pl" ucker relations. We prove that the coordinate ring of ${\text{Fl}}_{\lambda }^a$ is isomorphic to a direct sum of dual PBW-graded $\text{g}$-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.