Equivariant Embeddings of Algebraic Groups

equivalent to one-parameter subgroups γ 1 ′ , γ′ 2 ∈ X ∗(T ) and δ i = [γ i ′]. By part 4, γ′ 1 , γ′ 2 ∈ Γ(X, x 0)
as well. Then γ 1 ′ , γ′ 2 ∈ Γ(X, x 0) ∩ X ∗ (T ), which is the strongly convex rational polyhedral cone
associated to the toric variety T ⊂ X by part 3. As strongly convex rational polyhedral cones are
convex, the line in X ∗ (T ) joining γ 1 and γ 2 is contained in Γ(X, x 0 ) ∩ X ∗ (T ), and hence the line
in the spherical building joining δ 1 and δ 2 is contained in the image of Γ(X, x 0 ), so this image is
semi-convex. It is convex by part 2, which implies no pair of antipodal points of the building can
belong to the image of Γ(X, x 0 ).
Consider the following examples.
Example 5 (G viewed as an affine G-embedding). Consider the trivial G-embedding, G ⊂ G
with base point e. Here Γ(G, e) = {γ ∈ X ∗ (G) : lim t→0 γ(t) exists in G} = {ε}, where ε denotes
the trivial one-parameter subgroup, t ↦→ e, of G.
Example 6 (One variety as two distinct embeddings). Let B = {( a −1 b
0 a
)
: a, b ∈ k, a ≠ 0
}
,
which is not a reductive group. Then B → X = A 2 via ( a −1 b
0 a
)
↦→ (b, a) is a B-embedding. Here
⎧ ⎛ ⎞ ⎫
⎪⎨
⎪⎬
Γ(X, (0, 1)) =
⎪⎩ γ(t) = ⎜
⎝ t−n 0 ⎟
⎠ : n ∈ N 0 .
0 t n ⎪ ⎭
However, B → Y = A 2 given by ( a −1 b
0 a
)
↦→ (a −1 , a −1 b) is another B-embedding into A 2 , but
⎧ ⎛
⎞
⎫
⎪⎨
⎪⎬
Γ(Y, (1, 0)) =
⎪⎩ γ(t) = ⎜
⎝ t−n c(t n − t −n ) ⎟
⎠ : c ∈ k, −n ∈ N 0 .
0 t n ⎪ ⎭
Therefore, isomorphism of embeddings is a finer relation than isomorphism of varieties. As a result,
our notation Γ(X, x 0 ) may seem a bit clumsy as it does not indicate the manner in which B is
embedded in X, but this will be clear from the context whenever it is used.
Example 7 (Γ(X, x 0 ) for the group embedding associated to a toric variety). Suppose T is
a maximal torus of G and T σ is an affine toric variety for T . Then G × T T σ is an affine embedding
of G. Clearly σ ⊆ Γ(G × T T σ , [e, e]), as σ = Γ(G × T T σ , [e, e]) ∩ X ∗ (T ). For each parabolic
subgroup P of G containing T , let ∆ P (T ) = {γ ∈ X ∗ (T ) : P (γ) ⊇ P }. Then Γ(G × T T σ , [e, e]) =
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