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
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
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 |
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
(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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