Equivariant Embeddings of Algebraic Groups

G/B × G/B − corresponding to the B × B − -representation on the module H q (X, S| X ) ⊗ k M [e,e] .
Now S| X is a line bundle on the toric variety X, so S| X = O X (D h ) for some Σ-linear support
function h = h S .
First consider the B ×B − -representation on M [e,e] . Since M is a line bundle on the flag variety
G/B × G/B − ∼ = G × G/B × B − , it corresponds to a pair of characters (ξ, η) ∈ X ∗ (T ) × X ∗ (T ) [21].
Therefore B × B − ’s representation on M [e,e] corresponds to the one-dimensional representation of
B × B − on k associated to the character (ξ, η): (b 1 , b 2 ) · a = ξ(b 1 )aη(b −1
2 ), where ξ, η ∈ X∗ (T ).
Therefore the B ×B − -action on M [e,e] effectively twists its action on H q (X, S| X ), which we discuss
next.
Since B × B − acts on X via the map β : (b 1 , b 2 ) ↦→ β(b 1 ) and T ’s action on X, we obtain
a B × B − -representation in the cohomology groups H q (X, S| X ) for all integers q ≥ 0.
Let us
first note that S| X = i ∗ S is a line bundle on the toric variety X, and so is of the form O X (D h )
for some Σ-linear support function h. (Since X is complete, Pic(X) ∼ = SF (X ∗ (T ), Σ).) As a T -
representation, H q (X, O X (D h )) is a direct sum of its T -eigenspaces H q (X, O X (D h )) λ , λ ∈ X ∗ (T ).
Since the B × B − -action on X factors through that of T , this decomposition also holds as a
B × B − -representation. Thus, we have
⎡
H q (X, S| X ) ⊗ k M [e,e] = ⎣
⊕
λ∈X ∗ (T )
= ⊕
λ∈X ∗ (T )
= ⊕
λ∈X ∗ (T )
H q Z(h S ,λ) (X ∗(T ) R , k)e λ ⎤
⎦ ⊗ k M [e,e]
[H q Z(h S ,λ) (X ∗(T ) R , k)e λ ⊗ k M [e,e]
]
[
H q Z(h S ,λ) (X ∗(T ) R , k)e (λ+ξ,η)] .
To compute the sheaf H q (X, L), we use the following.
Lemma 6 ([26], Propositions I.3.3, I.5.9). If V, W are two K-representations, where K is a
closed subgroup of H, then I H/K (V ⊕ W ) = I H/K (V ) ⊕ I H/K (W ).
Proof. I H/K (V ⊕ W ) is the sheaf U ↦→ [O H (π −1 U) ⊗ (V ⊕ W )] K , where U is an open set in H/K
and π : H → H/K is the natural projection. Now [O H (π −1 U) ⊗ (V ⊕ W )] K = [(O H (π −1 U) ⊗
V ) ⊕ (O H (π −1 U) ⊗ W )] K = (O H (π −1 U) ⊗ V ) K ⊕ (O H (π −1 U) ⊗ W ) K , which is exactly (I H/K (V ) ⊕
I H/K (W ))(U).
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