Equivariant Embeddings of Algebraic Groups
Equivariant Embeddings of Algebraic Groups
Equivariant Embeddings of Algebraic Groups
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
Lemma 12. If γ 1 , γ 2 ∈ X ∗ (T ) ⊂ X ∗ (G), then<br />
(k[G] ∩ O vγ1 ) ∩ (k[G] ∩ O vγ2 ) ⊆ k[G] ∩ O vγ1 +γ 2<br />
, (3.17)<br />
where O vγ = {f ∈ k(G) : v γ (f) ≥ 0} is the valuation ring associated to v γ .<br />
Pro<strong>of</strong>. Recall that v γ is the valuation v t ◦ i γ <strong>of</strong> k(G), where i γ : k(G) → k(G)((t)) is the injection<br />
<strong>of</strong> fields filling the commutative diagram<br />
k[G]<br />
⊂<br />
k(G)<br />
µ ◦ <br />
k[G] ⊗ k[G]<br />
i γ<br />
id k[G] ⊗γ ◦ k[G] ⊗ k((t))<br />
⊂<br />
k(G)((t))<br />
and v t : k(G)((t)) × → Z is the standard valuation. Suppose γ 1 , γ 2 ∈ X ∗ (T ) ⊂ X ∗ (G) are oneparameter<br />
subgroups <strong>of</strong> G. Then γ 1 + γ 2 ∈ X ∗ (T ) is also a one-parameter subgroup <strong>of</strong> G contained<br />
in T . The valuations v γ1 , v γ2 and v γ1 +γ 2<br />
are obtained as above from the homomorphisms i γ1 , i γ2<br />
and i γ1 +γ 2<br />
, so it is enough to prove that i γ1 +γ 2<br />
(k[G] ∩ O vγ1 ∩ O vγ2 ) ⊂ k(G)[[t]] to prove the lemma.<br />
Yet<br />
k[G] ∩ O vγi<br />
k[G] ⊗ k[[t]]<br />
⊂<br />
k[G]<br />
µ ◦ k[G] ⊗ k[G]<br />
id k[G] ⊗γ ◦ i<br />
⊂<br />
k[G] ⊗ k((t))<br />
for i = 1, 2 and<br />
k[G]<br />
µ ◦ k[G] ⊗ k[G]<br />
id k[G] ⊗µ ◦ <br />
k[G] ⊗ k[G] ⊗ k[G]<br />
id k[G] ⊗(γ 1 +γ 2 ) ◦ <br />
id k[G] ⊗γ ◦ 1 ⊗γ◦ 2<br />
k[G] ⊗ k((t))<br />
id k[G] ⊗m 23<br />
k[G] ⊗ k((t)) ⊗ k((t))<br />
55