Equivariant Embeddings of Algebraic Groups

implies that
i γ1 +γ 2
(k[G] ∩ O vγ1 ∩ O vγ2 ) = (id k[G] ⊗ (γ 1 + γ 2 ) ◦ ) ◦ µ ◦ (k[G] ∩ O vγ1 ∩ O vγ2 )
= (id k[G] ⊗ [m 23 ◦ (γ ◦ 1 ⊗ γ ◦ 2) ◦ µ ◦ ]) ◦ µ ◦ (k[G] ∩ O vγ1 ∩ O vγ2 )
⊂ k[G] ⊗ k[[t]].
Thus, (k[G] ∩ O vγ1 ) ∩ (k[G] ∩ O vγ2 ) ⊆ k[G] ∩ O vγ1 +γ 2
as claimed.
Theorem 12. Affine G-embeddings are classified, up to conjugation, by strongly convex lattice
cones of one-parameter subgroups in X ∗ (G). Conjugation corresponds to the change of base point
in the embedding.
Proof. Assume X is an affine G-embedding.
The selection of a base point x 0 in X uniquely
determines a set Γ(X, x 0 ) ⊂ X ∗ (G), which is a strongly convex lattice cone by Lemma 11. By
Theorem 10, the set Γ(X, x 0 ) determines the affine G-embedding X and the selected base point x 0
via an isomorphism ψ x0 : k[X] ∼ = A Γ(X,x0 ) such that m x0 = ψ −1
x 0
(m e ∩ A Γ(X,x0 )). By Corollary 3
and formula (3.7), the selection of a different base point h · x 0 is equivalent to conjugating the
cone Γ(X, h · x 0 ) = hΓ(X, x 0 )h −1 and a corresponding right translation of the algebra A Γ(X,h·x0 ) =
r h (A Γ(X,x0 )).
Thus each affine G-embedding determines a strongly convex lattice cone, modulo
conjugation in X ∗ (G), which in turn recovers the variety up to isomorphism.
Conversely, we prove that every strongly convex lattice cone Γ ⊂ X ∗ (G) determines an affine
G-embedding X Γ such that Γ(X Γ , x 0 ) = Γ for some choice of base point x 0 ∈ X Γ . Assume Γ is a
strongly convex lattice cone, so Γ ∨ is a monoid state generated by a Kempf substate Ξ. Define A Γ
as in (3.9) to be the subalgebra of k[G] given by
A Γ = {f ∈ k[G] : v γ (f) ≥ 0, for all γ ∈ Γ} = k[G] ∩ ⋂ γ∈Γ
O vγ ,
where O v is the valuation subring {f ∈ k(G) : v(f) ≥ 0} of k(G). Using this second description,
since k[G] and all of the valuation rings O vγ
are integrally closed in k(G), their intersection A Γ is
an integrally closed domain. Furthermore, A Γ is left-invariant, since k[G] is and v γ (s · f) = v γ (f)
for all f ∈ k(G) and s ∈ G implies that the valuation rings O vγ
are G-stable as well. Hence A Γ is
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