Equivariant Embeddings of Algebraic Groups

one-dimensional orbits through (1, 0) and (0, 1) ∈ A 2 and the zero-dimensional orbit (0, 0). Under
the morphism GL 2 × D 2
A 2 → M 2 , these orbits are sent to GL 2 , ( a c 0
0 ) , ( )
0 b
0 d , (
0 0
0 0 ) in M 2. Therefore,
while it is clear that D 2
∼ = A 2 is the same in both GL 2 × D 2
A 2 and M 2 , these GL 2 -embeddings are
not isomorphic. For example, M 2 contains at least the fifth orbit through ( 1 1 1
1
), in addition to the
four in the image of GL 2 × D 2
A 2 . Therefore, there is no isomorphism between GL 2 × D 2
A 2 and M 2
as GL 2 -embeddings. Another point of interest which distinguishes the one-sided and biequivariant
cases is that with only a one-sided action, it is possible that there are maximal tori T 1 , T 2 such
that T 1
≁ = T 2 . This is obviously impossible in the presence of both a left and a right action, as all
maximal tori are conjugate in G.
While we cannot always recover our G-embedding X from T , we can associate to T other group
embeddings related to the original X in a number of ways. This is the topic of the next section.
2.3 Group embeddings from toric varieties
2.3.1 Mumford’s group embeddings
Not all group embeddings will respect both the left and right translations of G on itself, as is
demonstrated by the following construction due to Mumford [28]. He constructs an embedding
G ⊂ Ĝ of a semisimple group G into a reduced, irreducible and separated scheme Ĝ locally of finite
type over k satisfying the following conditions:
1. Ĝ is a toroidal embedding and, if G has no center, Ĝ is non-singular;
2. The left action of G on itself extends to an action of G on
Ĝ: i.e., there is a morphism
α : G × Ĝ → Ĝ extending the left multiplication in G;
3. The right action of G on itself extends pointwise but not continuously to an action on Ĝ: i.e.,
for all g ∈ G k , if R g : G → G is right multiplication by g, then R g extends to a morphism
̂R g : Ĝ → Ĝ;
4. For each stratum Y of Ĝ − G, {g ∈ G k : ̂R g (Y ) = Y } is the set of closed points of a parabolic
subgroup P Y of G and Y ↦→ P Y sets up a bijection between the strata of Ĝ − G and the
parabolics P ⊂ G.
22