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The Luna-Vust theory of spherical embeddings. (English) Zbl 0812.20023
Ramanan, S. (ed.), Proceedings of the Hyderabad conference on algebraic groups held at the School of Mathematics and Computer/Information Sciences of the University of Hyderabad, India, December 1989. Madras: Manoj Prakashan. 225-249 (1991).
Let $$k$$ be an algebraically closed field and let $$G$$ be a connected reductive group over $$k$$. The homogeneous variety $$G/H$$ is spherical if and only if a Borel subgroup of $$G$$ has an open orbit in $$G/H$$. Well- known examples for spherical homogeneous varieties are tori and symmetric varieties. A spherical embedding of $$G/H$$ is a normal $$G$$-variety $$X$$ together with a $$G$$-equivariant open embedding $$G/H \longrightarrow X$$.
Some 15 years ago Luna and Vust developed a method to classify embeddings (assuming $$\text{char} (k) = 0$$) by “coloured fans” (certain combinatorial data). Since then many results on spherical embeddings have been obtained.
In this well-written article the classification of spherical embeddings and several further results are presented. The author gives complete proofs, they are characteristic-free and most of them are shorter than the known ones.
For the entire collection see [Zbl 0777.00047].

MSC:
 20G05 Representation theory for linear algebraic groups 14L30 Group actions on varieties or schemes (quotients) 14M17 Homogeneous spaces and generalizations