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Unipotent commutative group actions on flag varieties and nilpotent multiplications. (English) Zbl 1393.14047
Consider the commutative unipotent group \((\mathbb{G}_a)^n\) over the field of complex numbers \(\mathbb{C}\). It is an important problem to study equivariant compactifications of the group \((\mathbb{G}_a)^n\). In other words, we are interested in actions with an open orbit of the group \((\mathbb{G}_a)^n\) on complete \(n\)-dimensional algebraic varieties \(X\).
In [B. Hassett and Y. Tschinkel, Int. Math. Res. Not. 1999, No. 22, 1211–1230 (1999; Zbl 0966.14033)], it was shown that equivariant compactifications of \((\mathbb{G}_a)^n\) with \(X\) being a projective space \(\mathbb{P}^n\) are in bijection with commutative associative local algebras over \(\mathbb{C}\) of dimension \(n+1\); see also [F. Knop and H. Lange, Math. Ann. 267, 555–571 (1984; Zbl 0544.14028)]. In particular, starting from \(n=6\) the number of equivalence classes of such compactifications is infinite.
Hassett and Tschinkel [Zbl 0966.14033] asked the same question for \(X\) being a non-degenerate projective quadric. By [E. V. Sharoǐko, Sb. Math. 200, No. 11, 1715–1729 (2009; Zbl 1205.13030); translation from Mat. Sb. 200, No. 11, 145–160 (2009)], in this case an equivariant compactification of \((\mathbb{G}_a)^n\) exists and is unique.
Let \(G\) be a semisimple complex linear algebraic group, \(P\) a parabolic subgroup of \(G\), and \(X=G/P\) the corresponding homogeneous space. Such varieties \(X\) are called generalized flag varieties, they are known to be the only complete homogeneous spaces of linear algebraic groups. In [I. V. Arzhantsev, Proc. Am. Math. Soc. 139, No. 3, 783–786 (2011; Zbl 1217.14032)], all homogeneous spaces \(G/P\) that admit an action with open orbit of the group \((\mathbb{G}_a)^n\) are found, and the question on the uniqueness of such an action is raised. In [B. Fu and J.-M. Hwang, Math. Res. Lett. 21, No. 1, 121–125 (2014; Zbl 1327.32030)], the uniqueness result is proved for a wide class of projective varieties including the Grassmanians \(\text{Gr}(k,m)\) different from projective spaces. The latter are precisely the varieties of the form \(G/P\) with \(G=\text{SL}(m)\) that admit an action of the group \((\mathbb{G}_a)^n\) with an open orbit.
In the paper under review, the uniqueness result is obtained for all generalized flag varieties \(G/P\), which are different from projectvie spaces and admit an action of the group \((\mathbb{G}_a)^n\) with an open orbit. The author establishes a correspondence between such actions and nilpotent multiplications on the nilpotent radical of the corresponding parabolic Lie subalgebra considered as an \(L\)-module with respect to the adjoint action of the Levi subgroup \(L\) of the parabolic subgroup \(P\).
Let \(V\) be a finite-dimensional module of a reductive algebraic group \(L\). One says that a bilinear map \(V\times V\to V, (v,w)\mapsto v\cdot w\), is an \(L\)-compatible nilpotent multiplication if this map is commutative, associative, the operator of multiplication \(V\to V, w\mapsto v\cdot w\) by any element \(v\in V\) is nilpotent and coincides with the operator \(V\to V, w\mapsto xw\), for some \(x\) in the Lie algebra of the group \(L\).
In Theorem 21, a classification of \(L\)-compatible nilpotent multiplications on simple modules \(V\) for a simple algebraic group \(L\) is obtained. This classification leads to the uniqueness result (Theorem 25).
For uniqueness results for non-commutative unipotent group actions with an open orbit on generalized flag varieties, see [D. Cheong, Transform. Groups 22, No. 1, 163–186 (2017; Zbl 1454.14126)].

MSC:
14M15 Grassmannians, Schubert varieties, flag manifolds
14L30 Group actions on varieties or schemes (quotients)
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References:
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