Equivariant Embeddings of Algebraic Groups

G. The wonderful compactification of G is the closure of the (G × G)-orbit of diag(g ⊕ g) in the
Grassmannian Grass dim G (g ⊕ g).
Example 2 (The wonderful compactication of P GL n ). For G = P GL n , P(M n ) is its wonderful
compactification.
Example 3 (A spherical compactification of SL 2 ). The group SL 2 has a nice biequivariant
compactification, namely the quadric in P 4 , Q = {[M : t] ∈ P(M 2 (k) × k) : det M = t 2 } with
SL 2 → Q given by M ↦→ [M : 1] and the action is by left and right multiplications on the first
coordinate, (g 1 , g 2 ) · [M : t] = [g 1 Mg −1
2 : t]. This is a spherical compactification of SL 2 .
In [5], Brion gave a combinatorial classification of regular compactifications [3] of semi-simple
groups in terms subdivisions of a Weyl chamber for the group, which is simpler than the Luna–Vust
classification for regular compactifications as complexity zero (G×G)-varieties. In general, however,
the complexity of G as a G-variety is too large (i.e., c(G) > 1) for the Luna–Vust approach to be
effective and, without a right G-action, Brion’s classification doesn’t apply. Thus the systematic
study of all equivariant group embeddings proposed by Danilov has been left virtually undone for
the past twenty five years.
1.4 Affine group embeddings
We study normal varieties X with a left G-action which possess an open subset isomorphic to an
algebraic group G so that the action of G on itself by (left) translations extends to the action of
G on all of X. As noted in [30], the combinatorial machinery of Luna and Vust cannot be easily
applied in this situation (as the complexity will be larger than 1 for most groups G). In its place,
we use the structure theory of the group G, and in particular its set of one-parameter subgroups,
X ∗ (G), to reduce the problem to classifying combinatorial data that we naturally attach to an
embedding of the group.
Let G be a connected reductive algebraic group defined over an algebraically closed field k. By a
G-variety, we mean a k-variety X together with a morphism ρ : G × X → X, written (g, x) ↦→ g · x,
satisfying g 1 · (g 2 · x) = (g 1 g 2 ) · x and e · x = x for all g 1 , g 2 ∈ G and all x ∈ X, where e ∈ G denotes
the identity element. We are interested in the following class of G-varieties.
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