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