By David C. Vella
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13) more carefully. 13) is a desirable result to have. 13) is not generally true. Of course, in addition to any vanishing behavior we do find, we should also be concerned with the structure of those L n J ( M J ( - X )) which do not vanish. Our results in this direction are deferred until after we develop some machinery involving the CPS subgroups. 1 of [12] yields that L - ^ j W U 2 Lp nL L (M) and Pj fl Lj is just the parabolic subgroup of Lj containing B A Lj and determined by I. Since both sides of this isomorphism can only be zero simultaneously, we may as well concentrate on Lp ^ L (M) .
0. The main ones involve, under a similar identification to the above, statements about the cohomology groups of vector bundles on PJ/PJ, not just line bundles. Although the notion of ampleness has been extended to vector bundles in [18], the nice cohomological properties of ample line bundles are not very well reflected in ample vector bundles when the characteristic is positive. 10. 13) turns out not to be true in 32 DAVID C. VELLA general when 1 * 0 . 11) Let X be a character of Pj, with X e A+.
7) (a) Hn(Pj,V) S Hn(B,v|B) for all V e Rat(Pj). ) This immediately generalizes to the case 1 * 0 : (b) Hn(Pj,V) s Hn(PI,v|p ) for all V € Rat(Pj) since by (a), each side is isomorphic to H n (B,V| B ). and I c J, There is another proof in [13], stated for J = A which is valid in this generality. That paper also contains a number of useful results in %%2 and 3 which have relative versions. 7). 4) to get relative versions. Similar care must be exercised in extending %3 of [13]. 8) Let X e A+ be a weight and V a finite dimensional rational Pj-module such that X is not strictly less than - Y for any weight Y of iP- V.