Equivariant Embeddings of Algebraic Groups

from X with a line bundle from the flag variety G/B × G/B − .
Corollary 2. There is an exact sequence of Picard groups associated to our fibration construction
of Ind(X):
0 Pic(G/B × G/B − )
p ∗ Pic(Ind(X)) i∗ Pic(X) 0.
Moreover, there is a section s : Pic(X) → Pic(Ind(X)) of i ∗ given by [L] ↦→ [(µ ∗ π ∗ 3 L)B×B− ].
Proof. We first show that p ∗ is injective.
Suppose p ∗ [M] = p ∗ [N ] for line bundles M, N on
G/B × G/B − . Then there is an isomorphism p ∗ M → p ∗ N . This pushes down to an isomorphism
p ∗ p ∗ M → p ∗ p ∗ N . However, if L is a line bundle on G/B × G/B − , then p ∗ p ∗ L = L: by definition
p ∗ p ∗ L is the sheaf associated to the presheaf U ↦→ p ∗ L(p −1 U) = lim −→V ⊃p(p
L(V ) = L(U), since
−1 U)
p is surjective implies p(p −1 U) = U. Therefore, since L is already a sheaf, p ∗ p ∗ L = L. Hence p ∗ is
an injection.
We now show that the sequence is exact at Pic(Ind(X)). First suppose that M is a line bundle
on G/B ×G/B − . Then i ∗ (p ∗ M) = (p◦i) ∗ M is (p◦i) −1 M⊗ (p◦i) −1 O G/B×G/B − O X, where (p◦i) −1 M
is the sheafification of U ↦→ lim −→V ⊃(p◦i)(U)
M(V ) = lim −→V ∋[e,e]
M(V ) = lim −→V ⊂U[e,e]
k = k. So i ∗ (p ∗ M)
is the constant sheaf k tensored with the structure sheaf of X, which is isomorphic to the structure
sheaf of X, O X . Hence the image of p ∗ is contained in the kernel of i ∗ . Moreover, if L is a line
bundle on Ind(X) such that i ∗ L is trivial, then the restriction of L to the fibers of p is always
trivial. Therefore p ∗ L is a line bundle on G/B × G/B − whose pull-back p ∗ p ∗ L is isomorphic to L
by the adjointness of p ∗ and p ∗ . Hence the sequence is exact at Pic(Ind(X)).
Finally, i ∗ is surjective, since we have already seen that s : [L] ↦→ [(µ ∗ π ∗ 3 i∗ L) B×B− ] is a section
satisfying i ∗ ◦ s = 1.
Since X = X Σ is a complete toric variety for T , its equivariant Picard group corresponds with
the set of Σ-linear support functions on X ∗ (T ). Moreover, the Picard group of the flag variety
G/B × G/B − is well-known to be isomorphic to the character group of the maximal torus T × T
of G × G [21], namely X ∗ (T ) × X ∗ (T ). Therefore,
Pic(Ind(X)) ∼ = SF (X ∗ (T ), Σ) ⋉ (X ∗ (T ) × X ∗ (T )), (2.2)
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