Equivariant Embeddings of Algebraic Groups

Lemma 12. If γ 1 , γ 2 ∈ X ∗ (T ) ⊂ X ∗ (G), then
(k[G] ∩ O vγ1 ) ∩ (k[G] ∩ O vγ2 ) ⊆ k[G] ∩ O vγ1 +γ 2
, (3.17)
where O vγ = {f ∈ k(G) : v γ (f) ≥ 0} is the valuation ring associated to v γ .
Proof. Recall that v γ is the valuation v t ◦ i γ of k(G), where i γ : k(G) → k(G)((t)) is the injection
of fields filling the commutative diagram
k[G]
⊂
k(G)
µ ◦
k[G] ⊗ k[G]
i γ
id k[G] ⊗γ ◦ k[G] ⊗ k((t))
⊂
k(G)((t))
and v t : k(G)((t)) × → Z is the standard valuation. Suppose γ 1 , γ 2 ∈ X ∗ (T ) ⊂ X ∗ (G) are oneparameter
subgroups of G. Then γ 1 + γ 2 ∈ X ∗ (T ) is also a one-parameter subgroup of G contained
in T . The valuations v γ1 , v γ2 and v γ1 +γ 2
are obtained as above from the homomorphisms i γ1 , i γ2
and i γ1 +γ 2
, so it is enough to prove that i γ1 +γ 2
(k[G] ∩ O vγ1 ∩ O vγ2 ) ⊂ k(G)[[t]] to prove the lemma.
Yet
k[G] ∩ O vγi
k[G] ⊗ k[[t]]
⊂
k[G]
µ ◦ k[G] ⊗ k[G]
id k[G] ⊗γ ◦ i
⊂
k[G] ⊗ k((t))
for i = 1, 2 and
k[G]
µ ◦ k[G] ⊗ k[G]
id k[G] ⊗µ ◦
k[G] ⊗ k[G] ⊗ k[G]
id k[G] ⊗(γ 1 +γ 2 ) ◦
id k[G] ⊗γ ◦ 1 ⊗γ◦ 2
k[G] ⊗ k((t))
id k[G] ⊗m 23
k[G] ⊗ k((t)) ⊗ k((t))
55