Let G be a semisimple algebraic group over an algebraically close d fiel d
k. For any closed subgroup H of G there i s a lef t exact functor "induction"
taking a rational H-module V t o a rational G-module V G. Induction and i t s
derived functors L|J
( _ ) provide a way of comparing th e representation theory
for G with that of H.
The cas e when H = B i s a Borel subgroup ha s been intensel y studied.
However, B i s th e smallest of th e parabolic subgroups, and i t may prove
expedient t o study th e entir e family of parabolic subgroups. Many properties of
Lg Q(_J carry over t o Lp
(_). However, counterexamples ar e given t o show that
Kempf's vanishing theorem does not extend t o thi s case.
One way t o study Lp
(V) i s t o restric t i t t o another parabolic subgroup
Q. When n = 0, there ar e known necessar y and sufficien t conditions on P and Q
to guarantee that v | ^ L * V ° |
where H = (P) °nQ. We extend thi s t o derived
functors, showing that L P G ( V M O ~ LH 0^V H ^ ^ o r n + 1 less than an integer
depending on H.
Effective use of this theorem requires understanding the functor (__) I g.
W e show that (_) j § imitates (_) | g in that it takes irreducible modules to
indecomposable modules and it preserves finite dimensionality. Furthermore, in
some cases when H is solvable (mainly involving the special linear group) we
show that XI § has a good filtration (with sections induced from a Borel
subgroup lying between H and Q). In particular \ | g is acyclic for the Levi
factor of Q.
Finally we apply these techniques when G is SL3(k) and M is a two
dimensional irreducible P module with negative dominant low weight. W e obtain
the vanishing behavior of all such Lp G(M) and we give dimension formulas for
M|£ and for Lj^G(M).
1980 Mathematics Subject Classification: 20G05 20J05 14F05
Key words: rational representations, induced modules, vector bundles on G/P
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