Equivariant Embeddings of Algebraic Groups
Equivariant Embeddings of Algebraic Groups
Equivariant Embeddings of Algebraic Groups
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Let X ∗ (G) denote the set <strong>of</strong> one-parameter subgroups <strong>of</strong> G. The group G acts on X ∗ (G) by<br />
conjugation, g • γ : t ↦→ gγ(t)g −1 . We will denote the trivial one-parameter subgroup t ↦→ e by ε.<br />
Each one-parameter subgroup γ ∈ X ∗ (G) determines a subgroup<br />
P (γ) = {g ∈ G : γ(t)gγ(t −1 ) ∈ G k[[t]] } (3.3)<br />
<strong>of</strong> G, which is parabolic if G is reductive [32]. In fact, every parabolic subgroup <strong>of</strong> a reductive<br />
group G is <strong>of</strong> the form P (γ) for some one-parameter subgroup γ <strong>of</strong> G [40]. We define an equivalence<br />
relation on the set <strong>of</strong> non-trivial one-parameter subgroups <strong>of</strong> G by<br />
γ 1 ∼ γ 2 if and only if γ 2 (t n 2<br />
) = gγ 1 (t n 1<br />
)g −1 (3.4)<br />
for positive integers n 1 , n 2 and an element g ∈ P (γ 1 ), for all t ∈ k × . Then the quotient (X ∗ (G) −<br />
{ε})/ ∼ is isomorphic to the spherical building <strong>of</strong> G [32], [48].<br />
Every parabolic subgroup P <strong>of</strong> G defines a subset ∆ P (G) = {γ ∈ X ∗ (G) : P (γ) ⊇ P } <strong>of</strong><br />
X ∗ (G). Clearly γ ∈ ∆ P (γ) (G) for all γ ∈ X ∗ (G), so X ∗ (G) = ⋃ ∆ P (G) where the union is over<br />
all parabolic subgroups <strong>of</strong> G. In the spherical building <strong>of</strong> G, the images <strong>of</strong> the sets ∆ P (G) are<br />
simplices and constitute a “triangulation” <strong>of</strong> the building so that it is the complex formed from<br />
these simplices [32].<br />
Given an affine G-embedding X with a choice <strong>of</strong> base point x 0 ∈ Ω, let<br />
Γ(X, x 0 ) = {γ ∈ X ∗ (G) : lim<br />
t→0<br />
γ(t)x 0 exists in X}. (3.5)<br />
Before we proceed, we make some immediate observations about such sets.<br />
Proposition 7. Let G be a connected reductive group. Suppose X is an affine G-embedding and<br />
x 0 ∈ X is a base point.<br />
1. If x ′ 0 = hx 0, then Γ(X, x ′ 0 ) = hΓ(X, x 0)h −1 .<br />
2. If γ ∈ Γ(X, x 0 ) and γ ≠ ε, then γ −1 ∉ Γ(X, x 0 ).<br />
3. If T is any torus <strong>of</strong> G, then T x 0<br />
∼ = T σ , where σ = Γ(X, x 0 ) ∩ X ∗ (T ) is a strongly convex<br />
40