Equivariant Embeddings of Algebraic Groups

since G (and hence G × G) is semi-simple, the short exact sequences
0 → H p λ
(N i−1 ) → H p λ
(N i ) → H p λ
(L i ) → 0
split, implying that H p λ(N i ) = H p λ(N i−1 ) ⊕ H p λ(L i ) = H p λ(L 1 ) ⊕ H p λ(L 2 ) ⊕ · · · ⊕ H p λ(L i ), by
induction, for i = 1, 2, . . . , r. In particular, H p λ(N r ) = ⊕ r
i=1 Hp λ(L i ), so that
H p (G/B × G/B − ,I G×G/B×B −(H q (X, S| X ) λ ⊗ k M [e,e] )) = H p (N r )
⎧
⎪⎨ H p λ(G/B × G/B − , L(λ + ξ, η)) ⊕ dim Hq Z(h,λ) (X∗(T ) R,k)
p = p λ
=
⎪⎩ 0 p ≠ p λ
Therefore,
E pq
2 = Hp (G/B × G/B − , H q (X, L))
= ⊕ [
] H p λ
(G/B × G/B − , L(λ + ξ, η)) ⊕ dim Hq Z(h,λ) (X∗(T ) R,k)
.
{λ:p λ =p}
Since these cohomology groups will be (G × G)-representations, they are also T -representations via
T → T × T → G × G. Furthermore, all the maps must respect this T -action, so a λ-eigenspace may
only be sent to another λ-eigenspace. However, the λ-eigenspaces occur in at most one column of
the spectral sequence E pq
2 , so all boundary maps must be zero. Therefore the sequence is degenerate,
so we may compute the cohomology of Ind(X) as the abutment:
H n (Ind(X), L) ∼ = ⊕
p+q=n
∼= ⊕
p+q=n
E pq
2
⎛
⎝
⊕
{λ:p λ =p}
[
H p λ
(G/B × G/B − , L(λ + ξ, η)) ⊕ dim Hq Z(h,λ) (X∗(T ) R,k) ]⎞ ⎠ .
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