2 DAVID C. VELLA
Also note that because induction preserve s the injectivit y of a module,
transitivit y (1.2) leads to a spectra l sequence of composite functors (see [19])
called the spectral sequence of induction, where for V e Rat(H):
(1.5)
E^'q(V)
2
LKfG(LH,K(V))f a n d c o n v e r
9
e s
to a filtratio n of
Lg+g(V).
To effectivel y us e induction a s a way of comparing G-modules with
H-modules, we should obtain a s much information a s i s possibl e about I»2 G(V) for
al l n, not jus t for n = 0. It i s convenient to be able to realiz e thes e G-modules
in other guises . For example (see [13]), we may identify Vljj with k[G] $
H
V,
where k[G] i s the coordinate ring of G. Then i t i s not difficult to show that
L[J G(V) identifie s with the rational cohomology group
Hn(H,k[G]
$ V). Another way
(see [10] or [3]) i s to construct from V the induced bundle L(V) on the quotient
space G/H. Then the global section s r(G/H/L(V)) form a rational G-module
isomorphic to V . Denote the quotient map G -• G/H by *. If * i s locall y trivia l
then V •* L(V) i s an exact functor taking infectiv e modules to acyclic sheaves ,
so i t follows that LS
G
(V) identifie s naturally with the sheaf cohomology group
Hn(G/H,L(V)). The techniques of algebraic geometry may then be applied.
For example, if H = P i s a parabolic subgroup, then * i s locall y trivia l
and moreover, G/P i s a projective variety. In particular, if V e Rat(P) i s finit e
dimensional then Lp
G
(V) i s finit e dimensional for al l n £ 0, and i s zero when
n dim G/P. (See Chapter 3 of [17].) Moreover, Serre duality may be applied in
t h i s situation. When H = B i s a Borel subgroup of G, then there are further
simplifications. B's irreducible representation s are a l l one dimensional, with U
acting triviall y and T acting vi a some character X in A. We will denote the
representation by X a s well. (Remark: We will usuall y assume G i s simply
connected s o that the character group X(T) i s al l of A, the ful l weight lattice.)
In thi s cas e the sheave s L(X) are al l rank one, that is , they are lin e bundles.
(For example, se e [3].)
The cas e H = B has been intensel y studied. As an indication of the
importance of thi s case, we remind the reader of Bott's theorem in
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