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