Equivariant Embeddings of Algebraic Groups

Theorem 9 ([27], Theorem 1.4). Let X be an affine G-variety. Suppose that Y is a closed
G-stable subvariety of X and that x 0 ∈ X is a closed point such that the closure of the orbit Gx 0
intersects Y . Then there is a one-parameter subgroup γ of G such that lim t→0 γ(t)x 0 ∈ Y .
Proof. Let y be a k-point of Y , which is contained in the closure of Gx 0 . We may find a curve η in
Gx 0 that has y in its closure. Take a curve ξ in G that dominates η under the morphism g ↦→ g · x 0 .
Let p : C → G be the rational mapping from a smooth complete curve C, which represents ξ. By
construction, there is a k-point c 0 of C such that lim c→c0 p(c) · x 0 = y.
Let R = Spec k[[t]] be the spectrum of the formal power series ring in one variable t. As the
completion of the local ring O C,c0
of the curve C at c 0 is isomorphic to k[[t]], we have a rational
mapping q : R → G such that lim t→0 q(t) · x 0 = y, where 0 is the closed point (t = 0) of R.
By Theorem 8, we may find two morphisms α 1 , α 2 : R → G such that α 1 · q = 〈γ〉 · α 2 , where
〈γ〉 : R → G is the the rational mapping given by taking the Laurent series expansion at 0 of a
one-parameter subgroup γ of G.
Let g i be the k-point α i (0) of G. The following limits exist and are equal in X by Lemma 8:
g 1 · y = lim
t→0
[α 1 (t)] · lim
t→0
[q(t) · x 0 ] = lim
t→0
[〈γ〉(t) · α 2 (t) · x 0 ].
Unfortunately, this limit does not always equal lim t→0 [γ(t) · g 2 · x 0 ]. However, the results of the
following claim are sufficient to complete the proof.
Claim ([27]). 1. The limit lim t→0 [γ(t) · g 2 · x 0 ] exists in X.
2. If X is a representation of G and y is the zero point 0, then lim t→0 [γ(t) · g 2 · x 0 ] = 0.
First, we will show how the claim implies the theorem. By Lemma 7, we may find a G-
equivariant morphism f : X → W , where W is a G-representation and Y = f −1 (0). By part 1 of
the claim, z := lim t→0 [γ(t)·g 2·x 0 ] exists in X. To show that z is in Y , it is enough to prove f(z) = 0,
where f(z) = lim t→0 [γ(t) · g 2 · f(x 0 )]. As the above argument applies equally well for x ′ = f(x 0 ),
y ′ = f(y) = 0 and X ′ = W , by part 2 of the claim, we must have f(z) = lim t→0 [γ(t) · g 2 · x ′ ] = 0.
Therefore, lim t→0 [g −1
2 γ(t)g 2 · x 0 ] = g −1
2 · z exists and is contained in Y . Taking λ(t) = g −1
2 γ(t)g 2,
we have found the desired one-parameter subgroup.
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