Equivariant Embeddings of Algebraic Groups

X ∗ (G).
Furthermore, [(Γ G ) ∨ ] ∨ = {γ ∈ X ∗ (G) : µ(G ∗ Ξ, γ) ≥ 0} = {γ ∈ X ∗ (G) : µ(g ∗ Ξ, γ) ≥ 0 for all g ∈
G} = ⋂ g∈G {γ ∈ X ∗(G) : µ(g ∗ Ξ, γ) ≥ 0} = ⋂ g∈G gΓg−1 = Γ G . Thus, Γ G is a convex lattice cone.
Finally, as Γ G ⊂ Γ, it is clear that γ, γ −1 ∈ Γ G implies that γ = ε is the trivial one-parameter
subgroup of G. Furthermore, ε ∈ hΓh −1 for all h ∈ G, so ε ∈ Γ G . Thus γ, γ −1 ∈ Γ G if and only if
γ = ε. Therefore, Γ G is a strongly convex lattice cone in X ∗ (G).
Definition 12. Let X be an affine G-embedding and select a base point x 0 ∈ X. Let Γ = Γ(X, x 0 )
and define Γ G = ⋂ h∈G hΓh−1 , which is a strongly convex lattice cone.
Then X G := Spec A Γ G
is a (G × G)-equivariant affine G-embedding by Theorem 12 and Proposition 10, and there is a
left-G-equivariant morphism β (X,x0 ) : X G → X corresponding to the inclusion A Γ(X,x0 ) ⊂ A Γ G
of
subalgebras in k[G]. We call X G together with the left-G-equivariant morphism β (X,x0 ) : X G → X
the biequivariant resolution of X.
We make a few remarks about the biequivariant resolution of an affine G-embedding before we
prove its universal property. First, if X is already a (G × G)-equivariant affine G-embedding, then
Γ = Γ G , so X G = X. However, it is possible in some cases that X G = G even when X ≠ G.
Example 15 (A trivial biequivariant resolution). Let D 2 denote the maximal torus of diagonal
matrices in GL 2 . Let σ be the strongly convex lattice cone generated by γ 1,0 : t ↦→
( t 0
0 1 ) and let T σ be the corresponding toric variety for D 2 . Observe that T σ = Spec k[σ ∨ ] =
Spec k[χ (1,0) , χ (0,1) , χ (0,−1) ] ∼ = A 1 × G m , whose base point is (1, 1) ∈ A 1 × G m . Consider the affine
GL 2 -embedding X = GL 2 × D 2
T σ . Then Γ(X, [I 2 , (1, 1)]) = ⋃ P ⊃D 2
P • (σ ∩ ∆ P (D 2 )). There are
only three parabolic subgroups of GL 2 containing D 2 : B + = {( )}
a b
0 d , B − = {( )}
a 0
c d , and GL2 .
As we have shown in Example 7, ∆ B +(D 2 ) = {γ m,n ∈ X ∗ (D 2 ) : m ≥ n}, ∆ B −(D 2 ) = {γ m,n ∈
X ∗ (D 2 ) : m ≤ n}, and ∆ GL2 (D 2 ) = {γ m,n ∈ X ∗ (D 2 ) : m = n}, where γ m,n is the one-parameter
subgroup of D 2 , t ↦→ ( )
t m 0
0 t . n Therefore, Γ(X, [I2 , (1, 1)]) = B + • σ and Γ(X, [I 2 , (1, 1)]) GL 2
=
⋂
g∈GL 2
g(B + • σ)g −1 ⊆ B + • σ ∩ s(B + • σ)s −1 , where s = ( 0 1 1
0 ) ∈ GL 2. Now a typical element of
B + • σ has the form ( )
1 b
0 1 (
t n 0
0 1 ) ( ) ( )
1 −b
0 1 =
t n b(1−t n )
for a non-negative integer n and an element
0 1
( ) ( )
b ∈ k. An element of s(B + • σ)s −1 is of the form ( 0 1 1
0 ) ( 0 1 1
0 ) = 1 0
b(1−t m ) t m for a
t m b(1−t m )
0 1
non-negative integer m and an element b ∈ k. Thus B + • σ ∩ s(B + • σ)s −1 = {ε}, which implies
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