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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:
 [1] B. Hasset, Yu. Tschinkel, Geometry of equivariant compactifications of G_{$$a$$}\^{}{$$n$$}, Intern. Math. Research Notices 1999 (1999), no. 22, 1211-1230. [2] E. В. Шаройко, Соответствие Хассетта-Чинкеля и автоморфизмы квадрики, Мат. Сб. 200 (2009), no. 11, 145-160. English transl.: E. V. Sharoiko, Hassett-Tschinkel correspondence and automorphisms of the quadric, Sb.: Math. 200 (2009), no. 11, 1715-1729. · Zbl 1349.94004 [3] I. V. Arzhantsev, Flag varieties as equivariant compactifications of G_{$$a$$}\^{}{$$n$$}, Proc. Amer. Math. Soc. 139 (2011), no. 3, 783-786. · Zbl 1217.14032 [4] N. Bourbaki, Groupes et Algébres de Lie, Chaps. IV, V, VI, Hermann, Paris, 1968. Russian transl.: H. Бурбаки, Группы и алгебры Ли. Группы Кокстера и системы Титса. Группы, порожденные отражениями. Системы корней, Мир, М., 1972. [5] M. Demazure, Automorphismes et déformations des variétés de Borel, Invent. Math. 39 (1977), no. 2, 179-186 · Zbl 0406.14030 [6] C. P. Ramanujam, A note on automorphism groups of algebraic varieties, Math. Ann. 156 (1964), 25-33. · Zbl 0121.16103 [7] H. Matsumura, F. Oort, Representability of group functors, and automorphisms of algebraic schemes, Invent. Math. 4 (1967), 1-25. · Zbl 0173.22504 [8] A. L. Onishchik, Topology of Transitive Transformation Groups, Johann Ambrosius Barth., Leipzig, 1994. · Zbl 0796.57001 [9] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, 3rd ed., Springer, Berlin, 1980. Russian transl.: Дж Хамфрис Введение в теорию алгебр Ли и их представлений МЦНМО, М., 2003. · Zbl 0467.00010 [10] A. L. Onishchik, E. B. Vinberg, Lie Groups and Algebraic Groups, Springer-Verlag, Berlin, 1990. · Zbl 0722.22004 [11] [11]. Э. Б. Винберг, В. В. Горбацевич, А. Л. Онищик, Строение групп и алгебр Ли, в книге: Группы Ли и алгебры Ли—3, Итоги науки и техн., Совр. пробл. матем. Фунд. напр., T. 41, ВИНИТИ, М., 1990, Cтр. 5-257. Engl. transl.: A. L. Onishchik, E. B. Vinberg, V. V. Gorbatsevich, Structure of Lie groups and Lie algebras, in: Lie Groups and Lie Algebras III, Encyclopaedia of Mathematical Sciences, Vol. 41, Springer-Verlag, Berlin, 1994, pp. 3-248. · Zbl 1120.30301 [12] J. Kollár, Rational Curves on Algebraic Varieties, Springer-Verlag, Berlin, 1996.
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