Equivariant Embeddings of Algebraic Groups

complement of U in G/H.
Remark 1 ([30], page 220). The Luna–Vust classification strategy is only feasible when c(K) ≤ 1.
Let P be the set of f ∈ k(G) which are simultaneously eigenvectors of B (acting by left
translations) and of H (acting by right translations).
f = g ∏ D∈D B f v D(f)
D
, where g is some element of k[G] × .
Any f ∈ P may be written in the form
In order to state the classification result of [30] when c(G/H) = 0, we need two more ideas.
Let X ∗ (H) be the group of characters of H.
If f ∈ k(G) is an eigenfunction of H, let χ f
be its character so χ f ∈ X ∗ (H). For E ⊂ k(G), let X ∗ E (H) be the set of χ ∈ X∗ (H) such that
E χ ≠ 0, where E χ is the set of H-eigenvectors in E of character χ. In order to describe the final
requirement of Luna and Vust’s result, we focus on two such E’s. The first is E = k[G] and the
second is E = P(E) = {f ∈ P : v D (f) = 0, for all D ∈ E} for a subset E of D B .
Denote by V the finite dimensional Q-vector space obtained by tensoring with Q the free Z-
module (P ∩k(G/H))/G m . The elements of V and the v D , D ∈ D B , may be thought of as elements
in the dual vector space V ∗ . If E ⊂ D B and W ⊂ V, let C(W, E) be the convex cone in V ∗ generated
by W and {v D : D ∈ E}.
Proposition 1 ([30], Proposition 8.10). Let H be a closed subgroup of G such that c(G/H) = 0.
Let E ⊂ D B and let W be a finite subset of V. Then there exists a G/H-embedding X and a closed G-
subvariety Y of X such that W is the set of valuations corresponding to the irreducible components
of X − G/H which are of codimension 1 in X containing Y and E is the set of v D ∈ D B such that
the closure of D in X contains Y if and only if the following conditions are satisfied:
1. The cone C(W, E) is strongly convex.
2. The lines Qw (w ∈ W) are the extremal lines of C(W, E) and do not coincide with any of the
Qv D (D ∈ E).
3. C(W, E) 0 ∩ V ≠ ∅, where C 0 denotes the relative interior of a cone C.
4. The group of characters of H is generated, as a monoid, by X ∗ k[G] (H) and X∗ P(E) (H).
Moreover, in terms of cones and sets as above, a pair (X ′ , Y ′ ) is an open G-subvariety of (X, Y )
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