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].


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