Equivariant Embeddings of Algebraic Groups

toric varieties. As [B, B] is the unipotent radical U of B and B/U ∼ = T , there is a unique, welldefined
homomorphism β : B → T .
Through this homomorphism, every T -variety Z is also a
B-variety, where b · z := β(b)z for all b ∈ B and all z ∈ Z.
Now let B − denote the Borel subgroup of G which is opposite to B, so that B ∩ B − = T .
Define the homomorphism β ′ : B × B − → T by (b, c) ↦→ β(b). Via β ′ , any T -variety becomes a
(B × B − )-variety, so in particular every toric variety for T is also a (B × B − )-variety.
Suppose X = X Σ is a toric variety for T associated to the fan, Σ, in X ∗ (T ) R . Then X is a
(B × B − )-variety as described above, so define Ind(X) := (G × G) × B×B− X. That is, Ind(X) =
[(G×G)×X]/ ∼ where we define ((g 1 , g 2 ), x) ∼ ((g 1 b 1 , g 2 b 2 ), β ′ (b 1 , b 2 ) −1 x) for all (b 1 , b 2 ) ∈ B ×B − .
We call the variety Ind(X) the induced toric variety associated to X. Let [(g 1 , g 2 ), x] denote the
equivalence class of ((g 1 , g 2 ), x) in Ind(X). Then Ind(X) is a left (G × G)-variety via the left action
of G × G on the first component: (h 1 , h 2 ) · [(g 1 , g 2 ), x] = [(h 1 g 1 , h 2 g 2 ), x].
We wish to elaborate on this construction. Given a T -variety X, we define the induced variety
Ind ∗ (X) to be the principal fiber space over the flag variety G/B × G/B − ∼ = G × G/B × B − with
fiber X under the (B × B − )-action just described. That is, there is a surjection p : Ind ∗ (X) →
G/B × G/B − all of whose fibers are isomorphic to X, and we may embed X in Ind ∗ (X) as the
fiber over (eB, eB − ).
Lemma 4. Suppose X is a toric variety for T . Then Ind(X) ∼ = Ind ∗ (X) as (G × G)-varieties.
Proof. Let X be a toric variety for T . Then consider the (G × G)-variety Ind(X) and the map
π 1 : Ind(X) → G × G/B × B − given by [(g 1 , g 2 ), x] ↦→ [g 1 , g 2 ]. This is well-defined, for if
(g 1 , g 2 , x) ∼ (g 1 b 1 , g 2 b 2 , β ′ (b 1 , b 2 ) −1 x), then π 1 ([(g 1 b 1 , g 2 b 2 ), β ′ (b 1 , b 2 ) −1 x]) = [g 1 b 1 , g 2 b 2 ] = [g 1 , g 2 ] =
π 1 ([(g 1 , g 2 ), x]).
Moreover, π −1
1 ([g 1, g 2 ]) = {[(h 1 , h 2 ), y] ∈ Ind(X) : π([(h 1 , h 2 ), y]) = [g 1 , g 2 ]} =
{[(g 1 b 1 , g 2 b 2 ), y] ∈ Ind(X)} = {[(g 1 , g 2 ), β ′ (b 1 , b 2 ) −1 y]} ∼ = X, since y is arbitrary. Therefore, Ind(X)
is fiber space over G × G/B × B − whose fiber over each point is isomorphic to X. Thus using
the isomorphism between G × G/B × B − and G/B × G/B − , we obtain the desired isomorphism
Ind(X) ∼ = Ind ∗ (X), which is clearly (G × G)-equivariant.
From now on we will identify Ind(X) with Ind ∗ (X) and refer to both as the induced toric
variety associated to X. However, we will see that this second description of the induced variety is
of greater value when determining the isomorphism classes of line bundles on Ind(X).
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