Equivariant Embeddings of Algebraic Groups

the category of toric varieties to the category of (G × G)-varieties as claimed.
Via the diagonal map ∆ : G → G×G, we have a functor (( (G×G)-varieties )) → ((G-varieties )),
so it follows that
Corollary 1. X ↦→ Ind(X) is a covariant functor (( T -varieties )) → (( G-varieties )).
As an aside, we describe the cohomology groups of line bundles on Ind(X) when it is projective
in terms of the cohomology groups of the toric variety X and of the flag variety G/B × G/B − .
Therefore, we first recall how to compute cohomology for toric varieties and flag varieties.
Cohomology of Line Bundles over Toric Varieties: Suppose X = X Σ is a toric variety.
The equivariant Picard group of X Σ corresponds with the set of Σ-linear support functions on
X ∗ (T ): a function h : |Σ| → R is called a Σ-linear support function if it is Z-valued on X ∗ (T ) ∩ |Σ|
and is linear on each σ ∈ Σ, i.e., there are linear maps l σ ∈ X ∗ (T ) ∨ such that h(v) = 〈l σ , v〉 for all
v ∈ σ and 〈l σ , v〉 = 〈l τ , v〉 whenever v ∈ σ ∩ τ. We denote the set of Σ-linear support functions
SF (X ∗ (T ), Σ). Then SF (X ∗ (T ), Σ) ∼ = Z Σ(1) by h ↦→ (h(σ 1 )) σ∈Σ(1) , where Σ(1) = {σ ∈ Σ : dim σ =
1} and σ 1 denotes the unique element of the ray σ ∈ Σ(1) such that σ = {nσ 1 : n ∈ N 0 }. The Σ-
linear support functions h determine T -equivariant divisors D h = − ∑ σ∈Σ(1) h(σ 1)D σ on X Σ , where
D σ is the closure of the orbit through z σ = lim t→0 σ 1 (t). Every T -equivariant line bundle L on X
is thus of the form O X (D h ) for some Σ-linear support function h, so SF (X ∗ (T ), Σ) ∼ = Pic T (X Σ ).
Suppose X = X Σ is a complete toric variety (i.e., Σ is a complete fan in X ∗ (T ) R ). Then the
cohomology groups H p (X, O X (D h )) are T -representations via the action of T on X. As such, since
T is linearly reductive, they split into a direct sum of T -eigenspaces:
H p (X, O X (D h )) ∼ =
⊕
H p (X, O X (D h )) λ ,
λ∈X ∗ (T )
where λ runs over the set of characters of T . Now define Z(h, λ) to be the closed subset {n ∈
X ∗ (T ) R : 〈 u λ , n 〉 ≥ h(n)} ⊂ X ∗ (T ) R , where u λ ∈ X ∗ (T ) ∨ corresponds to the character λ ∈ X ∗ (T ).
Then we have the following result.
Proposition 5 ([12], Theorem 7.2). For each λ ∈ X ∗ (T ), the eigenspace H q (X, O X (D h )) λ of
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