Section 1. Introduction and Review of Induced Modules.

Let k be an algebraically closed fiel d of characteristi c p 0, and l e t G

be a semisimple algebraic group over k. The category Rat(G) of rational

G-modules i s abelian and has enough injectives . We shal l be concerned with

comparing rational G-modules with rational P-modules vi a the proces s of

induction, where P i s a parabolic subgroup of G. The reader can consult [6] and

[20] for parabolic subgroups and [8] for basic fact s about induction.

Briefly, l e t u s here recal l that if H i s any closed subgroup of G, then the

restrictio n functor V - V

H

from Rat(G) to Rat(H) i s an exact functor admitting a

(left exact) right adjoint called induction from H to G. For V e Rat(H), the

induced module i s denoted by V | § or just V

G

if H i s understood. The following

propertie s about (_) ^ are basic.

(1-1) Reciprocity: HomG(W,v|G) * HomH(w|H,V) for al l G-modules W and H-modules V.

(1.2) Transitivity: If H C K are close d subgroups of G then v|j^|£ 2 v | ^

for any rational H module V.

U'3) Tensor identity: For V e Rat(H) and W e Rat(G) we have an

isomorphism of G-modules (V $ W) I g ^ VI gj $ W.

For V € Rat(H), one defines v |

G

to be MapR(G,V) from which (1.1) follows

easily . (1.1) merely e x p r e s s e s the fact that restrictio n and induction are

adjoints. (1.2) and (1.3) are eas y consequences of (1.1). (See [8] for proofs.)

Because (__) jj i s not exact in general, we are lead to consider i t s right derived

functors, which we denote by LjJ Q(__). We remark that dimension-shifting allows

u s to transfer (1.3) to the derived functors:

(1.4) Lg

Q

(V 0 W) £ LpifG(V) & W f o r alJL n 0, if W e Rat(G).

Received by the editors May 23, 1984.

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