PARABOLIC SUBGROUPS AND INDUCTION 3

characteristic zero, which exhibits each irreducible G-module as a module

induced from B. When the characteristic is positive, the induced modules -X

are no longer irreducible, although their formal characters are the same as in

characteristic zero. The problem of determining the composition factors of -XI

is important because of its relation to the problem of a characteristic p

character formula. (See page 111 of [26].) As for the case n 0, we have the

vanishing theorem of Kempf [23] which says Lg

G

(-X) = 0 if X is a dominant

weight. Also see [13] for this and for the basic structure of -XI2.

Because of the large amount of information that can be extracted about

G-modules by studying induction from B to G, there is a tendency to ignore the

presumably more complete picture which may be obtained by considering the

entire family of parabolic subgroups in G. There is a strong connection between

Rat(P) and Rat(G). For example, consider Corollary 3.3 of [9] where it is shown

that a rational B-module V extends to G if and only if it extends to every

minimal parabolic subgroup containing B. Smith's theorem [28] also illustrates

this connection: Certain irreducible P-modules appear in irreducible G-modules

as spaces which are fixed by a suitable unipotent group. A proof of this can be

based on induction (see [26]) and we include a slight generalization of this in

15. We also mention Proposition 5.21 below in this connection, which gives a

relationship between the composition factors of -X S and those of -X 2, by

applying (_) 2 to a composition series of -X S. It is to be hoped that further

study of L2

Q

(_) will lead to other connections between P-modules and G-modules.

One way to study the structure of Lp

Q

(V) is to restrict it to another

subgroup. In particular if H and K are any closed subgroups of G and V is a

rational H-module, we could ask for information regarding the structure of v|jj|

K

.

Theorems which give this structure (usually in terms of modules induced up to

K) are generally called Mackey decomposition theorems. For algebraic groups,

the most general of these theorems appear in [12]. There are two results of that

paper which we use fairly often. The isomorphisms (4.1) and (4.2) below are

immediate consequences of Theorem 4.1 of [12], and are well known in any case.

Theorem 4.4 of [12] is a general Mackey decomposition theorem. In Example 4.5 of

[12] this theorem is specialized to parabolic subgroups, and so is specially