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

H£ 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