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$${\mathbb{G}_a^M}$$ degeneration of flag varieties. (English) Zbl 1267.14064
Let $$G$$ be a simple Lie group. The flag varieties associated with $$G$$ are the homogeneous spaces $$G/P$$ by the parabolic subgroups. These varieties can be realized as $$G$$-orbits $$G[v_{\lambda}]\subseteq\mathbb{P}(V_{\lambda})$$ in the projectivization of the simple $$G$$-modules $$V_{\lambda}$$ with the highest weight vector $$v_{\lambda}$$. Let us denote $$\mathcal{F}_{\lambda}=G[v_{\lambda}]$$.
In this paper, the author introduces a family of varieties $$\mathcal{F}_{\lambda}^a$$, which are flat degenerations of $$\mathcal{F}_{\lambda}$$. These varieties are equipped with an action of the $$M$$-fold product $$\mathbb{G}_a^M$$ of the additive group of the ground field with an open orbit, where $$M$$ is the dimension of a maximal unipotent subgroup of $$G$$. Moreover, there exists a larger group $$G^a$$ acting on $$\mathcal{F}_{\lambda}^a$$, which is a degeneration of the group $$G$$ and which 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 Grassmannians and thus into the product of projective spaces. The defining ideal of $$\mathcal{F}_{\lambda}^a$$ is generated by the set of degenerate Plücker relations. It is proved that the coordinate ring of $$\mathcal{F}_{\lambda}^a$$ is isomorphic to a direct sum of dual PBW-graded modules. Also it is shown that there exists bases in multi-dimensional components of the coordinate rings parametrized by the semistandard PBW-tableaux.

##### MSC:
 14L35 Classical groups (algebro-geometric aspects) 17B45 Lie algebras of linear algebraic groups
##### Keywords:
Lie group; flag variety; degeneration; Plücker relations
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##### References:
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