Equivariant Embeddings of Algebraic Groups

Moreover, Ind(X) is a G-embedding, for the image of diag(G × G) ∼ = G in Ind(X) is
G ∼ = G × T T
⊆ G × T X ∼ = p −1 (diag(G/B × G/B − )) ⊂ Ind(X)
open
open
since T is open in X and diag(G/B × G/B − ) ∼ = G/T is open in G/B × G/B − [21].
Using this description of the induced toric variety Ind(X), we can readily verify that induction
is a functor:
Lemma 5. Induction as defined above is a covariant functor
(( toric varieties/T )) → (( (G × G)-varieties )).
Proof. By the existence of fiber products, Ind(X) is a variety and we have already described its
(G × G)-action, so Ind maps toric varieties to (G × G)-varieties. Furthermore, if f : X 1 → X 2 is a
T -equivariant morphism of toric varieties, it determines a (G × G)-equivariant morphism Ind(f) :
Ind(X 1 ) → Ind(X 2 ) as follows. Consider the map id G×G × f : G × G × X 1 → G × G × X 2 :
(id G×G × f)(g 1 b 1 , g 2 b 2 , β ′ (b 1 , b 2 ) −1 x) = (g 1 b 1 , g 2 b 2 , f(β ′ (b 1 , b 2 ) −1 x))
= (g 1 b 1 , g 2 b 2 , β ′ (b 1 , b 2 ) −1 f(x))
∼ (g 1 , g 2 , f(x))
= (id G×G × f)(g 1 , g 2 , x)
for all (b 1 , b 2 ) ∈ B × B − and (g 1 , g 2 , x) ∈ G × G × X. Since Ind(X i ) = [(G × G) × X i ]/ ∼, it
follows that Ind(f) : Ind(X 1 ) → Ind(X 2 ) is well-defined. Thus induction is the functor given by
X ↦→ Ind(X) := (G×G)× B×B− X and sending the morphism f : X 1 → X 2 to Ind(f) : [(g 1 , g 2 ), x] ↦→
[(g 1 , g 2 ), f(x)].
Now we only have left to show that Ind(id X ) = id Ind(X) for all toric varieties X and that
Ind(g) ◦ Ind(f) = Ind(g ◦ f) whenever f : X 1 → X 2 and g : X 2 → X 3 are T -equivariant morphisms.
The first of these requirements is clear. The second follows since Ind(g ◦ f) is the map Ind(X 1 ) →
Ind(X 3 ) associated to the map id G×G × (g ◦ f) = (id G×G × g) ◦ (id G×G × f), the latter map
corresponding to the composition Ind(g) ◦ Ind(f). Therefore, induction is a covariant functor from
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