Equivariant Embeddings of Algebraic Groups

is over a finite set of integers J. Then γ(t)f(x 0 ) = ∑ t i w i , so lim t→0 γ(t)f(x 0 ) = 0 if and only if
i > 0 for all i ∈ J. But µ(Υ Y X,x 0
, γ) = min J, so Γ Y (X, x 0 ) = {γ ∈ Γ(X, x 0 ) : µ(Υ Y X,x 0
, γ) > 0}.
Given a Kempf state Ξ, define the set
Γ Ξ = {γ ∈ X ∗ (G) : µ(Ξ, γ) ≥ 0}. (3.15)
Example 11 (The Kempf state of M n as a GL n -embedding). Consider the GL n -embedding
into M n with base point I n . Here Ξ Mn,In
is the state which assigns
(
to the maximal torus D n of
r1 0
diagonal matrices in GL n the set {χ 11 , χ 22 , . . . , χ nn }, where χ jj
. .. = r j for j = 1, 2, . . . , n.
0 r n
)
This corresponds to the eigendecomposition E 11 + E 22 + · · · + E nn of I n relative to D n . The state
Ξ Mn,I n
is GL n -stable under the conjugation action (because M n is a biequivariant GL n -embedding,
see section 3.3.2 for more details), so Ξ Mn,I n
(gD n g −1 ) = g ! Ξ Mn,I n
(D n ) = {g ! χ 11 , . . . , g ! χ nn } for all
g ∈ GL n . The closed GL n -subvariety ∂M n = V (det) corresponds to the state Υ ∂Mn
M n,I n
to each torus the set {det}. Clearly
that assigns
Γ(M n , I n ) = {γ ∈ X ∗ (GL n ) : lim
t→0
γ(t)I n exists in M n }
= {γ ∈ X ∗ (GL n ) : min 〈χ, γ〉 ≥ 0}
χ∈Ξ Mn,In (γ(G m))
= {γ ∈ X ∗ (GL n ) : µ(Ξ Mn,I n
, γ) ≥ 0},
Γ ∂Mn (M n , I n ) = {γ ∈ Γ(M n , I n ) : lim
t→0
γ(t)I n exists in ∂M n }
= {γ ∈ Γ(M n , I n ) : 〈det, γ〉 > 0}
= {γ ∈ Γ(M n , I n ) : µ(Υ ∂Mn
M n,I n
, γ) > 0}.
Therefore, given an affine G-embedding X and a choice of base point x 0 ∈ Ω, we obtain a Kempf
state Ξ X,x0 such that Γ(X, x 0 ) = {γ ∈ X ∗ (G) : µ(Ξ X,x0 , γ) ≥ 0}. For each closed G-subvariety Y of
X, we obtain another Kempf state Υ Y X,x 0
such that Γ Y (X, x 0 ) = {γ ∈ Γ(X, x 0 ) : µ(Υ Y X,x 0
, γ) > 0}.
However, the Kempf state for (X, x 0 ) depends on the choice of G-representation V in Lemma 7.
Thus, for our purposes, rather than using Kempf states, we shall employ monoid states.
Definition 10. A monoid state is a state Ψ such that Ψ(R) is a monoid for each R and for which
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