Equivariant Embeddings of Algebraic Groups

that Γ(X, [I 2 , (1, 1)]) GL 2
= {ε} and that the biequivariant resolution of X is β (X,[I2 ,(1,1)]) : G → X
given by g ↦→ [g, (1, 1)].
Second, while the cone Γ G is canonical, the morphism β (X,x0 ) : X G → X is only defined
up to right translation, which corresponds to a different choice of base point x ∈ X as follows:
r h (β (X,x0 )) = β (X,h·x0 ) : X G → X. This is true because the identification of k[X] with a subalgebra
of k[G] is determined by the selection of a base point and changing the base point corresponds to
right translation of the subalgebra in k[G] by Corollary 3. Then the morphism β (X,h·x0 ) : X G → X
is defined by the inclusion of A Γ(X,h·x0 ) = r h (A Γ(X,x0 )) ⊂ A Γ G, from which it is clear that β (X,h·x0 ) =
r h (β (X,x0 )).
Proposition 11 (Universal Property of Biequivariant Resolutions). Suppose X is a left-
G-equivariant affine G-embedding, x 0 ∈ X is a base point, Y is any (G × G)-equivariant affine
G-embedding, and ϕ : Y → X is a left-G-equivariant morphism. Then there is a unique (G × G)-
equivariant morphism ϕ (XG ,x 0 ) : Y → X G such that ϕ = β (X,x0 )◦ϕ (XG ,x 0 ). That is, the biequivariant
resolution of X satisfies the following diagram:
Y
∀ ϕ
X
∃! ϕ (XG ,x 0 ) β (X,x0 )
X G
If x 1 = h · x 0 is another base point for X, then β (X,x1 ) = r h (β (X,x0 )) and ϕ (XG ,x 1 ) = r h −1(ϕ (XG ,x 0 )).
Proof. Let X be an affine G-embedding and identify k[X] with the left-invariant subalgebra A Γ(X,x0 )
of k[G] by selecting a base point x 0 ∈ X. Suppose that Y is a (G × G)-equivariant affine G-
embedding and that ϕ : Y → X is a left-G-equivariant morphism from Y to X. By equivariance,
there is a unique element y 0 ∈ Y such that ϕ(y 0 ) = x 0 , since ϕ| ΩY
: Ω Y → Ω X is an isomorphism
with Ω Y
∼ = ΩX ∼ = G. Then the cone Γ(Y, y0 ) is G-stable by Proposition 10 and is a subset of
Γ(X, x 0 ) by Proposition 9. Using y 0 , identify k[Y ] with the (G × G)-invariant subalgebra A Γ(Y,y0 )
of k[G]. By Corollary 3, A Γ(Y,h·y0 ) = r h (A Γ(Y,y0 )) = A Γ(Y,y0 ), for all h ∈ G, so the algebra A Γ(Y,y0 ) is
independent of the choice of base point. Hence it is the only subalgebra of k[G] which is isomorphic
to k[Y ] as a (G × G)-algebra by Theorem 12.
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