Equivariant Embeddings of Algebraic Groups

ational polyhedral cone in X ∗ (T ) (Section 2.1.1, Definition 3).
4. If γ ∈ Γ(X, x 0 ) and p ∈ P (γ), then pγ(t)p −1 ∈ Γ(X, x 0 ), and moreover
Γ(X, x 0 ) = ⋃
P • (Γ(X, x 0 ) ∩ ∆ P (G)) (3.6)
P ⊂G
where the union is taken over all parabolic subgroups P of G.
5. The image of Γ(X, x 0 ) in the spherical building is convex ([32], Definition 2.10).
Proof. First, Γ(X, x 0 ) depends on the base point x 0 as follows. Suppose x ′ 0 ∈ Ω, so x′ 0 = h · x 0
for some unique h ∈ G (because G → Ω is an isomorphism). Then Γ(X, x ′ 0 ) = hΓ(X, x 0)h −1 for
if γ ∈ Γ(X, x 0 ) (that is, if lim t→0 γ(t)x 0 ∈ X), then lim t→0 (hγ(t)h −1 )x ′ 0 = lim t→0 hγ(t)h −1 hx 0 =
lim t→0 hγ(t)x 0 = h lim t→0 γ(t)x 0 , which exists in X. Therefore hΓ(X, x 0 )h −1 ⊂ Γ(X, x ′ 0 ). By
symmetry, since x 0 = h −1 x ′ 0 , h−1 Γ(X, x ′ 0 )h ⊂ Γ(X, x 0), so Γ(X, x ′ 0 ) ⊂ hΓ(X, x 0)h −1 . Hence,
Γ(X, h · x 0 ) = h Γ(X, x 0 ) h −1 (3.7)
for any h ∈ G.
Second, as X is affine, if γ ∈ Γ(X, x 0 ) and γ is not the trivial one-parameter subgroup ε : t ↦→ e,
then γ −1 ∉ Γ(X, x 0 ). Otherwise, if both lim t→0 γ(t)x 0 and lim t→0 γ −1 (t)x 0 exist in X, then the
composition ψ x0 ◦ γ : G m → X extends to a morphism ˜γ : P 1 → X, which must therefore be
constant, so γ = ε.
Third, if T is any torus of G, then T x 0
∼ = Tσ , where σ ⊂ X ∗ (T ) is the strongly convex lattice
cone Γ(X, x 0 ) ∩ X ∗ (T ) by Theorem 3.
Now suppose γ ∈ Γ(X, x 0 ) and p ∈ P (γ).
Then p · γ · p −1 also belongs to Γ(X, x 0 ), for
lim t→0 [(pγ(t)p −1 )x 0 ] = lim t→0 [p(γ(t)p −1 γ(t −1 ))γ(t)x 0 ] = p[lim t→0 γ(t)p −1 γ(t −1 )][lim t→0 γ(t)x 0 ]
exists in X by Lemma 8 and the definition of P (γ). Therefore, it is clear that Γ(X, x 0 ) =
⋃
P ⊂G P • (Γ(X, x 0) ∩ ∆ P (G)), where the union is taken over all parabolic subgroups P of G.
Lastly, if δ 1 , δ 2 ∈ (Γ(X, x 0 )−{ε})/ ∼, then there are one-parameter subgroups γ 1 , γ 2 ∈ Γ(X, x 0 )
such that δ i = [γ i ] is the equivalence class of γ i . The one-parameter subgroups determine parabolic
subgroups P (γ 1 ) and P (γ 2 ), whose intersection contains a maximal torus T of G. Then γ 1 and γ 2 are
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