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

Abstract

Let $\Fl_λ$ 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 $\Fl_λ^a$, which is a ${\mathbb{G}}_a^M$ variety. Moreover, there exists a larger group $G^a$ acting on $\Fl_λ^a$, which is a degeneration of the group $G$. The group $G^a$ 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 Grassmanians and thus to the product of projective spaces. The defining ideal of $\Fl^a_λ$ is generated by the set of degenerate Pl\" ucker relations. We prove that the coordinate ring of $\Fl_λ^a$ 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.

24 pages

References in corpus (3)

• The PBW Filtration, Demazure Modules and Toroidal Current Algebras
• Zhu's algebras, $C_2$-algebras and abelian radicals
• Flag varieties as equivariant compactifications of G_a^n