Equivariant Embeddings of Algebraic Groups

2. Each G-orbit closure is the transverse intersection of the boundary divisors containing it.
3. If x ∈ X, then in the normal space to G · x in X the isotropy group G x of x acts with an
open orbit.
Definition 7. The regular compactifications of G are the biequivariant compactifications X of G
that are regular as (G × G)-varieties.
Example 4 (Some regular compactifications). Smooth complete toric varieties are regular
compactifications of tori. For an adjoint group G, De Concini and Procesi [7] have constructed the
wonderful compactification of G, as in Section 1.3. One could also define it as the unique regular
compactification of G which has a unique closed (G × G)-orbit, although the notion of regular
compactifications arose later. More recently, Brion [6] used the Hilbert scheme to recreate this
compactification. For example, the projective space P(M 2 (C)) is the wonderful compactification of
P GL 2 (C).
From now on, X will be a regular compactification of G. Fix a maximal torus T of G.
Let Φ(T, G) be the set of roots of G relative to T . Suppose Φ + is a set of positive roots of
Φ(T, G) and ∆ is its base. Let C + = {ν ∈ X ∗ (T ) R : ∀α ∈ ∆, 〈α, ν〉 ≥ 0}, the closure of the positive
Weyl chamber.
One now defines a subdivision of C + associated to X. For this, let T denote the closure of T in
X. On G, the restriction of the action of G×G to the torus diag(T ×T ) is given by (t, t)·g = tgt −1 .
This extends to all of X. The variety of fixed points for diag(T × T ) is smooth [23] and T is an
irreducible component. For the left action of T , T is thus a smooth complete toric variety, so one
has the associated fan Σ for T . As T is invariant by the diagonal action of the Weyl group, Σ is
also W -invariant. Hence Σ = W Σ + , where Σ + is the subdivision of C + formed of the cones of Σ
contained in C + .
In a manner analogous to the toric case, the cones of the fan Σ + parametrize the (G × G)-orbits
of X. For all σ ∈ Σ + , denote by z σ the corresponding base point in T and by O σ its (G × G)-orbit.
One says that z σ is the base point of the orbit O σ . The closed (G × G)-orbits of X correspond
to maximal dimensional cones of Σ + . When σ is such a cone, z σ has isotropy group B − × B, so
(G × G) · z σ
∼ = G/B − × G/B.
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