Equivariant Embeddings of Algebraic Groups

Mumford’s embedding is constructed as follows. For every Borel subgroup B ⊂ G, form ̂B in the
following way. First, let U be the unipotent radical of B. Second, let T = B/U and α i ∈ X ∗ (T ) the
roots of T in U. Third, let σ ⊂ X ∗ (T ) R be {x : 〈α i , x〉 ≥ 0, ∀i}. Fourth, split B = U × T . Finally,
form ̂B = B × T T σ . Only the fourth step is non-canonical and ̂B does not depend on the choice of
the splitting. With these ̂B’s constructed, define Ĝ = ⋃ B G ×B ̂B where the gluing is the unique
one so that all G × B ̂B’s, for all Borel subgroups B of G, are identified at least on their common
open set G ∼ = G × B B ⊂ G × B ̂B and Ĝ is separated and G × B ̂B ⊂ Ĝ is an open subset. Then
this embedding G ⊂ Ĝ satisfies the conditions above. In particular, Ĝ has only a left G-action, so
it does not fall within the scope of the spherical approach of Sections 1.2, 1.3, 2.2.1.
2.3.2 Group embeddings constructed using flag varieties
Let G be a semi-simple simply connected algebraic group with maximal torus T and Borel subgroup
B ⊃ T .
In Section 2.2, we saw how B-embeddings B ⊂ Y induced G-embeddings G ⊂ G × B Y via
g ↦→ [g, y 0 ]. If Y 1 → Y 2 is a B-equivariant morphism of B-embeddings, then there is a corresponding
G-morphism G× B Y 1 → G× B Y 2 . Moreover, if X is any G-embedding and B denotes the closure of
B in X, then there is an equivariant morphism G× B B → X of G-embeddings. If X is projective, so
that B is too, then G× B B is projective and the morphism G× B B → X is surjective. Furthermore,
if X 1 → X 2 is a morphism of G-embeddings, then, setting Y i = B ⊂ X i and constructing G × B Y i ,
we obtain a G-morphism G × B Y 1 → G × B Y 2 over X 1 → X 2 :
G × B Y 1
X 1
G × B Y 2
X 2 .
Therefore, the category of B-embeddings may be viewed as a full subcategory of the category of
G-embeddings by the functor Y ↦→ G × B Y , which is adjoint to the restriction functor X ↦→ B ⊂ X
from the category of G-embeddings to the category of B-embeddings. These results are true for
arbitrary closed subgroups H of G [36].
In this section, we give another construction of G-embeddings obtained using flag varieties and
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