PARABOLIC SUBGROUPS AND INDUCTION
5
H^MV)) all vanish for i d, as L(V) is coherent. Next, there is the spectral
sequence of local cohomology with E?J'q = HP(X/H^(L(V)) and converging to
Hp+q(X,L(V)). The above result gives HP(X,L(V)) = 0 for p d, which when
combined with the long exact sequence of local cohomology:
0 - H°(X,L(V)) - H°(X,I(V)) - H°(Q,L(V)|Q) - H|(X,L(V)) - .. . ,
gives that H^X^KV)) s
Ei(Q,L(V)
\Q) = H^L/L^MV*)) for i d - 1. This gives
Theorem 4.4 and permits us to conclude the above decomposition theorem for two
parabolics. Indeed when G is connected and semisimple, fix a Borel subgroup B
and a corresponding base A of $. For proper subsets J and K of A, let Pj and
PK be the corresponding parabolic subgroups. Then PK has an open orbit Q in
G/Pj, namely the orbit of wQPj. It turns out that codim(G/P - Q) ) 2 if and
i PK,
v
only if J u K = A, so
the theorem applies to give vlp |
p
s V °|
H
J where
J wo
is the CPS subgroup (Pj) n PK- This result i s extended in S6, when the
*
above codimension is computed explicitly by examining how J and K overlap.
(But see the appendix of [31] for a proof which avoids group-schemes by
sticking to parabolic subgroups and i = 0.)
The organization of this work is as follows: In §2 we compute the Levi
decomposition of a CPS subgroup. In % 3 we parametrize the irreducible P modules
by their highest weights, as is the case for irreducible G-modules. In % 4 we
begin the study of induction by describing the structure of ~X|B/ and
recording some analogues to the results of SS1-3 of [13]. All of this is more or
less preliminary.
In 15 we continue the study of induction more generally by considering
the effect of LP 2
D
( ) on an irreducible PT-module M. The basic idea of this
r
P j - I
I
P
J l
P
T
section i s to apply (_)|p to a composition series of -X|
B
. This leads to the
l
•fundamental sequence1 of M, which among other things exhibits
M|PpJ
as a
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