Equivariant Embeddings of Algebraic Groups

an integrally closed left-invariant subalgebra of k[G]. It only remains to prove that A Γ is finitely
generated over k and that the variety X Γ = Spec A Γ contains G as an open orbit.
Let γ 1 , . . . , γ m be a set of representatives for the equivalence classes in Γ(1)/ ∼, which is a finite
set as Γ is a lattice cone. By Proposition 8 and Lemma 12,
A Γ = k[G] ∩ ⋂ γ∈Γ
O vγ = ⋂ γ∈Γ(k[G] ∩ O vγ ) =
m⋂
(k[G] ∩ O vγi )
since any γ ∈ Γ belongs to some X ∗ (T ), in which case it is a sum of elements from (Γ ∩ X ∗ (T ))(1).
Let f 1 , . . . , f n be a finite set of generators for the algebra k[G] over k, so k[G] = k[f 1 , . . . , f n ]. By
Gordan’s Lemma, the set
C = {(c l ) ∈ Z n : c 1 , . . . , c n ≥ 0,
i=1
n∑
c j v γi (f j ) ≥ 0 for all i = 1, . . . , m}
j=1
is finitely generated. Thus the “monomials” f c 1
1 f c 2
2 · · · f n
cn
in the generators f 1 , . . . , f n of k[G]
associated to the generators (c 1 , . . . , c n ) of C admit a finite set of generators of A Γ , as any element
of A Γ is a sum of such monomials.
Therefore X Γ = Spec A Γ is a normal affine G-variety with an open G-orbit, since A Γ ⊂ k[G]
implies f : G → X Γ is dominant, so f(G) is an open orbit in X Γ ([40], Theorem 1.9.5). It only
remains to show that f(G) is isomorphic to G as an orbit in X Γ . Consider, for each torus R of
G, the image of A Γ under the homomorphism k[G] → k[R]. By construction of A Γ = {f ∈ k[G] :
v γ (f) ≥ 0 for all γ ∈ Γ} and the decomposition A Γ = ⊕ χ∈Γ ∨ (R) AR χ , where A R χ = {f ∈ A : f(xr) =
χ(r)f(x) for all r ∈ R}, it is evident that the image of A Γ in k[R] is the k-monoid algebra k[Γ ∨ (R)].
Hence we have the following commutative diagrams:
⊂
A Γ
k[G]
X Γ
f
G
onto
k[Γ ∨ (R)]
⊂
onto
k[R] = k[X ∗ (R)]
closed
R Γ ∨ (R)
open
R
closed
Since Γ is strongly convex, Γ ∨ (R) is not contained in any hyperplane, so the monoid Γ ∨ (R) generates
X ∗ (R) as a group. This implies that R is isomorphic to an open subset of the closed subvariety
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