On the structure of the cohomology of line bundles on [formula omitted

Abstract

One of the most intriguing problems in characteristic p representation theory for semisimple algebraic groups is the problem of finding the formal characters of the irreducible modules. An equivalent problem is to find the composition factors of Weyl modules. It is a (non-trivial) fact that the Weyl modules occur as the top cohomology groups of certain line bundles on homogeneous spaces. One could therfore more generally ask for the composition factors of all such cohomology groups. References [2] and [3] deal with aspects of this question. In the present paper we use a method primarily due to J. C. Jantzen [ 151 to extend the results in the abovementioned works. Our main results are (i) a translation principle (Theorem 2.5 below) generalizing Jantzen’s theorem in [ 15’1, (ii) a proof of the Carter-Lusztig conjecture (see [6, p. 2391, compare also [3, Theorem 4.11 and [ 71) on the existence of intertwining homomorphisms between Weyl modules (Theorem 3.3 below) together with some related results on higher Ext,-groups, and (iii) a rather general vanishing theorem for cohomology groups of line bundles on homogeneous spaces (Theorem 4.1 below). As will be apparent from a comparison with [ 151, many of Jantzen’s arguments in the proof of his translation principle can be directly carried over to our situation. However, the fact that we deal with all cohomology groups (not just the highest) allows us sometimes to shorten the proofs. In particular this is, the case for the proof of Proposition 2.3 (in which we give the effect of the translation functor on irreducible modules). In an earlier preprint [4], we showed that the vanishing theorem for dominant line bundles on homogeneous spaces can be proved (for p big) using “translation arguments.” Shortly after that preprint was written, the author 151 (and, independently, W. Haboush [ 111) found an even simpler way of proving that result. In order to describe the vanishing behaviour of the cohomology of non-ample line bundles the method is, however, still useful. This is demonstrated in the last section of this paper, where we 245 0021.8693/81,‘070245-14$02.00/O