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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)
##### Keywords:
semisimple groups; unipotent group; flag varieties
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##### References:
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