Equivariant Embeddings of Algebraic Groups

type 〈γ〉, for some one-parameter subgroup γ of G. That is,
G k((t)) =
⋃
G k[[t]] 〈γ〉G k[[t]] (3.1)
γ∈X ∗(G)
Moreover, each double coset is represented by a unique dominant one-parameter subgroup.
This decomposition will be essential for replacing k((t))-points of G with one-parameter subgroups
in the proof of Theorem 9 below.
For a one-parameter subgroup γ of G, recall that, by definition, lim t→0 γ(t)x 0 exists in X if
γ : G m → G, when composed with ψ x0
: g ↦→ g · x 0 , extends to a morphism ˜γ : A 1 → X and
lim t→0 γ(t)x 0 is defined to be ˜γ(0). That is, the composition of ψ ◦ x 0
: k[X] → k[G] with γ ◦ : k[G] →
k[t, t −1 ] factors through k[t], and the limit lim t→0 γ(t)x 0 is the k-point of X corresponding to the
composite k[X] → k[t] → k sending t → 0. This is described by the diagrams:
G m ⊂
γ
A 1˜γ 0
k[X]
˜γ 0 ◦
G
ψx0
X k[t]
ψ ◦ x 0
k[G]
γ ◦
⊂ k[t, t −1 ].
Similarly, if λ is a k((t))-point of G, then lim t→0 λ(t)x 0 exists in X means λ ◦ | k[X] : k[X] → k[[t]].
The following lemma is used frequently hereafter.
Lemma 8. Suppose λ ∈ G k((t)) and α ∈ G k[[t]] , so that α has specialization α 0 ∈ G k . Let
X be an affine G-embedding with base point x 0 . Then lim t→0 [λ(t)x 0 ] exists in X if and only if
lim t→0 [α(t)λ(t)x 0 ] exists, in which case
lim [α(t)λ(t)x 0] = α 0 · lim[λ(t)x 0 ]. (3.2)
t→0 t→0
Proof. Since α has a specialization in G k , α corresponds to a ring homomorphism α ◦ : k[G] → k[[t]].
Now consider λ ∈ G k((t)) . Suppose lim t→0 λ(t)x 0 ∈ X. Then λ ◦ ◦ψ ◦ x 0
: k[X] → k[G] → k((t)) factors
through k[[t]] ⊂ k((t)). Moreover, (αλ) ◦ ◦ ψ x0
is given by the composition k[X] → k[G] ⊗ k k[X] →
k[[t]] ⊗ k k[[t]] → k[[t]] in the diagram below, so αλ ∈ X k[[t]] . Hence, lim t→0 [α(t)λ(t)x 0 ] exists in X,
and its value is given by composing (αλ) ◦ ◦ ψ ◦ x 0
with the map k[[t]] → k sending t ↦→ 0. However,
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