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