\({\mathbb{G}_a^M}\) degeneration of flag varieties. (English) Zbl 1267.14064

Feigin, Evgeny

Sel. Math., New Ser. 18, No. 3, 513-537 (2012).

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.

Reviewer: Ivan V. Arzhantsev (Moscow)

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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