Equivariant Embeddings of Algebraic Groups

As x 0 is not zero, Ξ V,x0 (R) is a nonempty subset of X ∗ (R) for each torus R of G. Furthermore,
if R 1 ⊂ R 2 is an inclusion of tori of G, then the eigendecomposition of x 0 with respect to R 2 is
also the eigendecomposition with respect to R 1 whose weights are the restrictions of characters
χ ∈ Ξ V,x0 (R 2 ) to R 1 , so the restriction property is true. Hence Ξ V,x0
is a state.
Let x 0 = ∑ v i be the eigendecomposition of x 0 with respect to a one-parameter subgroup γ of
G. Then γ(t)·x 0 = ∑ t i v i , where each v i ≠ 0 and i runs through a finite nonempty set I of integers,
since X ∗ (G m ) ∼ = Z. Suppose R is a torus such that γ ∈ X ∗ (R). Then I = {〈χ, γ〉 : χ ∈ Ξ V,x0 (R)}
by the definition of a state.
As µ(Ξ V,x0 , γ) = min I, lim t→0 γ(t) · x 0 exists in V if and only if
µ(Ξ V,x0 , γ) ≥ 0. Thus Γ(X, x 0 ) = Γ(V, x 0 ) = {γ ∈ X ∗ (G) : µ(Ξ V,x0 , γ) ≥ 0}.
It may be shown that Ξ V,x0 is a bounded, admissible state. First of all, for any torus R and any
vector v ∈ V , the set Ξ V,v (R) is contained in the set of R-weights of V , which is finite. Furthermore,
observe that ∑ g · v χ is the eigendecomposition of g · x 0 with respect to R whenever ∑ v χ is the
eigendecomposition of x 0 with respect to T = g −1 Rg, so
Ξ V,g·x0 (R) = g ! (Ξ V,x0 (g −1 Rg)) = g ∗ Ξ V,x0 (R) (3.14)
for all g ∈ G k . Thus, the union ⋃ g∈G k
g ∗ Ξ V,x0 (R) is contained in the finite set of R-weights of
V , and hence is finite. Therefore, Ξ V,x0
is bounded. Now let γ be any one-parameter subgroup
of G. Let p be a k-point of P (γ) = {g ∈ G : γ(t)gγ(t −1 ) ∈ G k[[t]] }, the parabolic subgroup of G
associated to γ. As P (γ) = P (p·γ ·p −1 ), it will be enough to prove that µ(Ξ V,x0 , γ) ≤ µ(Ξ V,x0 , p·γ ·
p −1 ). We may think of µ(Ξ V,x0 , γ) as the largest integer n such that lim t→0 t −n γ(t)x 0 exists in V .
Then lim t→0 t −n pγ(t)p −1 x 0 = p −1
0 = lim t→0 t −n p(γ(t)p −1 γ(t −1 ))γ(t)x 0 = p[lim t→0 γ(t)p −1 γ(t −1 )] ·
[lim t→0 t −n γ(t)x 0 ], which exists in V by Lemma 8. Therefore, µ(Ξ V,x0 , p · γ · p −1 ) is at least n, so
Ξ V,x0
is admissible, as claimed.
To define Υ Y X,x 0
, we use part 2 of Lemma 7, which says that there is an equivariant morphism
f : X → W , for some G-representation W , such that Y = f −1 (0). Set Υ Y X,x 0
equal to the
state of the point f(x 0 ) in the vector space W , defined by (3.13) above. Therefore, Υ Y X,x 0
is a
bounded admissible state. Moreover, γ ∈ Γ(X, x 0 ) implies that γ ∈ Γ(W, f(x 0 )), so lim t→0 γ(t)x 0
is contained in Y if and only if γ ∈ Γ(X, x 0 ) and lim t→0 γ(t)f(x 0 ) = 0 in W . Let f(x 0 ) = ∑ w i
be the eigendecomposition of f(x 0 ) with respect to the action of γ, where the w i ≠ 0 and the sum
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