Equivariant Embeddings of Algebraic Groups

cone as above, then each Γ ∩ X ∗ (T ) is one in the sense of toric geometry as defined in Section 2.1.1,
Definition 3. For if Γ is a strongly convex lattice cone, Γ = {γ ∈ X ∗ (G) : µ(Ξ, γ) ≥ 0} for a Kempf
state Ξ which generates Γ ∨ . Thus each Γ ∩ X ∗ (T ) is the intersection of finitely many half-spaces,
Γ ∩ X ∗ (T ) = ⋂ χ∈Ξ(T ) {v ∈ X ∗(T ) : 〈χ, v〉 ≥ 0}, so it is a convex lattice cone in X ∗ (T ). The strong
convexity follows as the same condition is required of Γ.
Lemma 11. If X is an affine G-embedding with base point x 0 ∈ Ω, then the set Γ(X, x 0 ) = {γ ∈
X ∗ (G) : lim t→0 γ(t)x 0 exists in X} is a strongly convex lattice cone.
Proof. Observe that if (X, x 0 ) is an affine G-embedding, then Γ(X, x 0 ) is a convex lattice cone by
Proposition 8, Theorem 11 and our discussion above. Therefore, it is a strongly convex lattice cone,
for we have shown that γ, γ −1 ∈ Γ(X, x 0 ) implies γ = ε and that ε ∈ Γ(X, x 0 ) in Proposition 7.
Example 13 (The monoid state for Γ(G, e)). The monoid state corresponding to the group
G as an affine G-embedding is equal to X ∗ , for Γ(G, e) = {ε} consists of only the trivial oneparameter
subgroup and every character of every torus of G has non-negative value when paired
with ε. A Kempf substate for X ∗ may be found by viewing G as a closed subgroup of some GL n ,
which in turn is a closed subvariety of M n+1 defined by the equations det[(X ij ) n i,j=1 ]X n+1,n+1 = 1
and X i,n+1 = X n+1,j = 0 for 1 ≤ i, j ≤ n. Notice that g ∈ GL n acts on M n+1 through matrix
( {( ) }
g 0
g 0
multiplication by
0 det(g)
), −1 under which GL n =
0 det(g) −1 : g ∈ GL n is invariant. Then
Ξ G,e = res GLn
G
ΞGLn M n+1 ,I n+1
, which is a Kempf state by Remark 3, for Ξ Mn+1 ,I n+1
is a Kempf state by
the proof of Theorem 11.
The strong convexity condition implies that the dual monoid state Γ ∨ takes values Γ ∨ (R) that
generate X ∗ (R) for all tori R. For if Γ is strongly convex, then (Γ ∩ X ∗ (R)) ∩ (−Γ ∩ X ∗ (R)) = {ε},
so (Γ ∩ X ∗ (R)) ∨ + (−Γ ∩ X ∗ (R)) ∨ = [(Γ ∩ X ∗ (R)) ∩ (−Γ ∩ X ∗ (R))] ∨ = X ∗ (R). Therefore, Γ ∨ (R) =
(Γ ∩ X ∗ (R)) ∨ generates X ∗ (R) as a group for all R.
3.2.3 Classification theorem
Let G be a connected reductive algebraic group defined over an algebraically closed field k of
characteristic zero. Let T be a maximal torus of G.
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