b DAVID C. VELLA
lPJ lPJ
submodule of -X
B
for suitable X. This shows that structure of M
p
is
somewhat similar to that of
-XlpJ
B
(compare Proposition 4.3 with Proposition 5.1).
The similarity is not as marked when n 0. In generalizing Proposition 4.6
(Kempf's Theorem) to the case I * 0, one would hope to be able to prove that
Lp G(M) = 0 for n 0 if M has a low weight which is negative dominant (see
K
(5.13) below). However, we show in %5 and again in 18 that this is not true and
only the weaker version (5.12) holds. So the vanishing behavior of LS
D
( ) is
more complicated than the case 1 = 0. This is one reason that it is desirable to
extend the Mackey decomposition theorem discussed above to the higher derived
functors of induction. As mentioned above, this is done in 16.
In 17 we consider another aspect of the Mackey decomposition theorem.
1PK
Since thi s theorem involve s the functor (_)
H
, where H i s a CPS subgroup, we
aim in thi s sectio n to get specifi c information about thi s functor. We show that,
lik e (_) S, i t take s irreducibles to indecomposables and take s finit e dimensional
H-modules to finit e dimensional P
K
-modules (this i s not true for al l LJJ
p
(__)).
Furthermore, in certain c a s e s where H i s solvable and G i s of type An, and
P K
always when G i s of type B2, we show
X|l
H
has a filtratio n with quotients of
l p K
the form u
B
, where BR i s a conjugate of B which l i v e s between H and PK. Such
a filtratio n i s commonly called a good filtration. In particular, thi s means
that XI ix i s acyclic as an LK-module, where LR i s the Levi factor of PR. When
*
wn
-X = X i s a character of Pj, then thi s resul t combined with the Mackey
decomposition theorem s a y s that -X
B
i s L
K
-acyclic, and has a good filtratio n
for LK. The reader i s advised t o be aware that in t h i s sectio n and again in 18,
we frequently work relativ e to a new base AK which c o n s i s t s of conjugates of A
by the long word wK of WR.
18 i s the las t section, and in i t we return to the study of !*£ G(M) where
M i s an irreducible P-module. The methods of 17 are applied, and when combined
with the fundamental sequence of 15, we are able to obtain the vanishing
behavior of L^ Q(M) when M i s two dimensional with a negative dominant low
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