Equivariant Embeddings of Algebraic Groups

where SF (X ∗ (T ), Σ) is the abelian group of Σ-linear support functions.
Cohomology of Line Bundles over an Induced Toric Variety: From now on, we will
assume that the variety X is a complete toric variety corresponding to the fan Σ in X ∗ (T ) ∼ = Z r .
To compute the cohomology groups H • (Ind(X), L) of Ind(X) with coefficients in a (G × G)-
equivariant line bundle L, we take advantage of the decomposition of line bundles as described
in Proposition 6. Since Ind(X) may be viewed as a fibration of G/B × G/B − with fiber X, this
cohomology may be computed using a Leray spectral sequence.
Theorem 7 ([18], Leray Spectral Sequence). Given the fiber bundle p : Ind(X) → G/B ×
G/B − , suppose L is a (G × G)-equivariant line bundle on Ind(X). Then, for each integer q ≥ 0,
define the sheaf H q (X, L) to be the sheaf on G/B × G/B − generated by the presheaf
U ↦→ H q (p −1 U, L).
Then there exists a spectral sequence such that
E pq
2 = Hp (G/B × G/B − , H q (X, L)),
and whose abutment E ∞ is the bigraded group associated to a suitable filtration of the graded group
H • (Ind(X), L).
Thus, to compute the cohomology groups H • (Ind(X), L) as we desire, we must first compute
the sheaves H q (X, L) for each integer q ≥ 0.
Consider the stalk of the sheaf H q (X, L) at the
image of the identity element in the flag variety, [e, e]. Take an affine trivializing cover {U} of
G/B × G/B − . Then
H q (X, L) [e,e] = inj lim
[e,e]∈U
= inj lim
[e,e]∈U
H q (p −1 U, L) = inj lim H q (X × U, S ⊗ OInd(X) p ∗ M)
[e,e]∈U
⊕
r∈Z
= H q (X, S| X ) ⊗ k M [e,e] ,
H r (X, S| X ) ⊗ k H q−r (U, M) = inj lim H q (X, S| X ) ⊗ k H 0 (U, M)
[e,e]∈U
using the Künneth formula. Therefore H q (X, L) is the induced (G × G)-sheaf over the flag variety
30