Equivariant Embeddings of Algebraic Groups

(i.e., O X ′ ,Y ′ is a localization of O X,Y ) if and only if C X ′ ,Y ′ is a face of C X,Y and E X ′ ,Y ′ = {D ∈
E X,Y : v D ∈ C X ′ ,Y ′}.
Therefore, to determine a G-model X is the same thing as to determine the set of its pairs
(V Y , DY B ) as Y varies over the closed G-subvarieties of X, as then we may glue cones together to
form colored fans (with “colors” being the sets D B Y
⊂ DB ) and in this way characterize G/Hembeddings
in general.
Varieties of complexity zero are called spherical. These include flag varieties (and horospherical
varieties in general), toric varieties, and symmetric varieties. The theory of these spaces can be
unified with the concept of spherical varieties.
For example, all three types of spaces have a
nice compactification theory, respectively due to Pauer (horospherical varieties); Demazure, Kempf
et al., Miyake–Oda, etc. (toric varieties); and De Concini–Procesi, etc. (symmetric varieties).
Spherical varieties have been extensively studied over the past twenty years, so Knop reworked
and improved the results of [30] in the special case of spherical varieties, extending the Luna–Vust
classification to positive characteristic [29]. There are a number of results concerning the spherical
subgroups of G, cohomology computations, local structure theorems, the B-orbit structure, and
relations to symplectic geometry. Here is just one example:
Proposition 2 ([30], Proposition 7.5). Suppose c(K) = 0. Then the number of G-orbits in any
G-model of K is finite. In fact, any such G-model can only have finitely many B-orbits.
1.3 The “wonderful compactification”
A partial extension of toric geometry to embeddings of reductive groups may be obtained by viewing
the group G as a homogeneous variety for G×G under left and right translations [6], [7], [8], [9], [46].
In this way, all G × G-equivariant embeddings of G are spherical, so may be classified by means of
the Luna–Vust theory described above. For instance, the wonderful compactification of an adjoint
group, introduced and studied by De Concini and Procesi [7], [8], [9], is a special case of a G × G-
spherical compactification of G.
The wonderful compactification of an adjoint group G is constructed as follows. For the adjoint
action of G × G in its Lie algebra g ⊕ g, the isotropy group of the diagonal diag(g ⊕ g) is equal to
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