Equivariant Embeddings of Algebraic Groups

i ≥ 0, 〈χ ij , γ (0,1) 〉 = j ≥ 0} = {χ ij ∈ X ∗ (G 2 m) : i, j ≥ 0}:
• • • • •
• • γ (0,1)
• •
X ∗ (G 2
m) : • • ◦
γ (1,0) •
• X ∗ (R) • • •
• • • • •
Now consider the subtorus R = diag(G 2 m) = {(r, r) : r ∈ G m }.
Clearly Γ ∩ X ∗ (R) = ∅, so
Γ ∨ (R) = {χ ∈ X ∗ (R) : 〈χ, γ〉 ≥ 0 for all γ ∈ Γ ∩ X ∗ (R) = ∅} = X ∗ (R). Yet the restriction of
Γ ∨ (G 2 m) to X ∗ (R) is {χ j : j ≥ 0} X ∗ (R), where χ j (r, r) = r j .
If Γ ∨ is a state, then it is canonical, which is an advantage over the Kempf state Ξ X,x0 associated
to an affine G-embedding (X, x 0 ) by Theorem 11. Moreover, it is obvious that each Γ ∨ (R) is a
monoid, but this does not ensure that Γ ∨ is a monoid state, according to our definition, which
requires the existence of a Kempf substate Ξ of Γ ∨ that generates each Γ ∨ (R).
However, by
Theorem 11, if Γ = Γ(X, x 0 ) for some affine G-variety, then Ξ X,x0
is such an underlying Kempf
substate. Hence Γ(X, x 0 ) ∨ is a monoid state. Moreover, Theorem 11 ensures that Γ(X, x 0 ) = {γ ∈
X ∗ (G) : µ(Ξ X,x0 , γ) ≥ 0} = (Γ(X, x 0 ) ∨ ) ∨ . Thus, the combined requirements that Γ ∨ be a monoid
state and that Γ = (Γ ∨ ) ∨ eliminates unwanted sets Γ, which leads to the following definition:
Definition 11. We call a subset Γ ⊂ X ∗ (G) a lattice cone of one-parameter subgroups of G if Γ is
saturated with respect to the equivalence relation (3.4) of one-parameter subgroups (i.e., if γ 1 ∼ γ 2
and γ 1 ∈ Γ, then γ 2 ∈ Γ) and the quotient Γ(1)/ ∼ of the one-skeleton of Γ (3.11) is a finite set. A
lattice cone Γ is called a convex lattice cone if Γ ∨ is a monoid state and Γ = (Γ ∨ ) ∨ . Additionally,
Γ is a strongly convex lattice cone if it is a convex lattice cone and γ, γ −1 ∈ Γ if and only if γ = ε
is the trivial one-parameter subgroup of G.
Our terminology is compatible with that of toric geometry. If Γ is a strongly convex lattice
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