Equivariant Embeddings of Algebraic Groups

and we see Γ(GL 2 × D 2
A 2 , [e, e]) = ⋃ P ⊃D 2
P • (σ ∩ ∆ P (D 2 )).
Example 8 (The affine GL n -embedding M n ). Consider G = GL n and X = M n , the affine
space of n × n matrices. Clearly M n is an affine GL n -embedding with the obvious left action. Any
( t p 1 0
one-parameter subgroup of G can be obtained from a one-parameter subgroup t ↦→ . .. ,
0 t pn )
for p 1 , . . . , p n ∈ Z, after suitable conjugation by some matrix A ∈ GL n . Such a one-parameter
( t p 1 0
)
subgroup A . .. A −1 has a limit in M n if and only if p 1 , . . . , p n ≥ 0. Thus Γ(M n , I n ) =
( 0 t pn
t p 1 0
{A . .. A −1 : A ∈ GL n and p 1 , . . . , p n ∈ N 0 }.
0 t pn )
Each γ ∈ X ∗ (G) may be viewed as a k((t))-point of G.
In [30], a G-stable valuation v λ is
associated to every λ ∈ G k((t)) in the following way.
As λ is a k((t))-point of G, we obtain a
dominant morphism
G × Spec k((t)) 1×λ G × G µ G.
This morphism induces an injection of fields i λ : k(G) → Frac(k(G) ⊗ k k((t))) → k(G)((t)). Then
v t ◦ i λ : k(G) × → Z is a valuation of k(G), where v t : k(G)((t)) × → Z is the standard valuation
associated to the order of t. We define v λ = 1 (v t ◦i λ ), where n λ ∈ Z is the largest positive number
n λ
such that (v t ◦ i λ )(k(G) × ) ⊂ n λ Z (except when λ = ε, in which case v ε (f) = 0 or ∞ as f(e) ≠ 0 or
= 0, respectively). This is G-stable by left translations, i.e., v λ (s · f) = v λ (f) for all s ∈ G, since
i λ is clearly equivariant and k(G)[[t]] is obviously stable for left translations by G in k(G)((t)). We
include some of the properties of these valuations that are proven in [30] in the following lemma.
Lemma 9 ([30]).
1. Let γ be a one-parameter subgroup of G. For each f ∈ k(G), there is an
open subset U ⊂ G, depending only on f, such that
v γ (f) = inf
s∈U v t(f(s · γ(t))) (3.8)
2. Let γ 1 , γ 2 be one-parameter subgroups of G. Then v γ1 = v γ2 if and only if γ 1 ∼ γ 2 .
Proof. Part 1 is Lemma 4.11.1 in [30], where U = {s ∈ G : f(s) ≠ 0}. The second part is the result
of Propositions 3.3 and 5.4 in [30].
The sets of one-parameter subgroups Γ(X, x 0 ) described in (3.5) are significant for the following
reasons.
The first result, which will serve as our foundation for the classification theorem in
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