Equivariant Embeddings of Algebraic Groups

Therefore, since H q (X, S| X ) = ⊕ λ∈X ∗ (T )H q (X, S| X ) λ , we have
H q (X, L) = I G×G/B×B −(H q (X, S| X ) ⊗ k M [e,e] )
⊕
= I G×G/B×B −( [H q (X, S| X ) λ ⊗ M [e,e] ])
= ⊕
λ∈X ∗ (T )
λ∈X ∗ (T )
I G×G/B×B −(H q (X, S| X ) λ ⊗ k M [e,e] ),
so E pq
2 = Hp (G/B ×G/B − , H q (X, L)) = H p (G×G/B ×B − , ⊕ λ∈X ∗ (T )I G×G/B×B −(H q (X, S| X ) λ ⊗ k
M [e,e] )) = ⊕ λ∈X ∗ (T )H p (G/B × G/B − , I G×G/B×B −(H q (X, S| X ) λ ⊗ k M [e,e] )). Thus it suffices to
compute each
H p (G/B × G/B − , I G×G/B×B −(H q (X, S| X ) λ ⊗ k M [e,e] ))
separately and then to add them together. Now H q (X, S| X ) λ = H q Z(h,λ) (X ∗(T ) R , k)e λ , where
S| X = O X (D h ) for some Σ-linear support function h, by Proposition 5. Since H q (X, S| X ) λ ⊗ k
M [e,e]
∼ = H
q
Z(h,λ) (X ∗(T ) R , k)e (λ+ξ,η) , where M [e,e]
∼ = k · (ξ, η) as a B × B − -representation, we have
a filtration
H q Z(h,λ) (X ∗(T ) R , k)e (λ+ξ,η) = N r ⊃ N r−1 ⊃ · · · ⊃ N 2 ⊃ N 1 ⊃ N 0 = 0
where each L i = N i /N i−1 is a one-dimensional representation of B × B − of character (λ + ξ, η) and
r = dim H q Z(h,λ) (X∗ (T ) R , k). Hence we get short exact sequences
0 → N i → N i+1 → L i+1 → 0,
which yield long exact sequences of cohomology (where we suppress the G/B × G/B − in the
notation)
0 → H 0 (N i ) → H 0 (N i+1 ) → H 0 (L i+1 ) → H 1 (N i ) → H 1 (N i+1 ) → H 1 (L i+1 ) → · · ·
for i = 1, 2, . . . , r − 1. Now N 1 = L 1 , L 2 , L 3 , . . . , L r are line bundles over G/B × G/B − all of weight
(λ + ξ, η), so their cohomology groups are non-zero in at most one index, which we call p λ . Then,
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