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Flag varieties as equivariant compactifications of \(\mathbb{G}_{a}^{n}\). (English) Zbl 1217.14032
The author classifies all (generalized) flag varieties \(G/P\) which can be realized as an equivariant compactification of the commutative unipotent affine algebraic group \(\mathbb{G}_a^n\), i.e. \(\mathbb{G}_a^n\) acts on \(G/P\) with an open orbit. Here, \(G\) is a connected semisimple linear group of adjoint type over an algebraically closed field of characteristic zero and \(P\) is a parabolic subgroup of \(G\).
Note that, given any \(G/P\), the unipotent radical \(P_u^-\) of the opposite of \(P\) acts with an open orbit on \(G/P\). Moreover, \(P_u^-\) and such an orbit are affine spaces; in particular they are isomorphic to \(\mathbb{G}_a^n\) as varieties (with \(n=\dim \,G/P\)). So when \(P_u^-\) is commutative, it is isomorphic to \(\mathbb{G}_a^n\) and the desired action exists.
Suppose now that \(G\) is simple. There is another natural class of flag varieties with an action of \(\mathbb{G}_a^n\) as desired. Indeed, there are some \(G/P\) such that \(G\) is strictly contained in the connected component \(\widetilde{G}\) of the automorphism group of \(G/P\). If we write \(G/P\cong \widetilde{G}/\widetilde{P}\), then \(\widetilde{P}^-_u\) is commutative while \(P^-_u\) is not. So, \(G/P\) has an action of \(\mathbb{G}_a^n\) with an open orbit but \(\mathbb{G}_a^n\) is not isomorphic to \(P^-_u\).
The author proves that the only \(G/P\) with an action of \(\mathbb{G}_a^n\) as requested are products of flag varieties in the previous two classes. Finally, for fixed \(G/P\), there can exist infinitely many equivalence classes of actions of \(\mathbb{G}_a^n\) on \(G/P\) with an open orbit.

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