On the intertwinings between higher cohomology modules of line bundles on [formula omitted

Abstract

Let G be a simple, simply connected affine algebraic group over an algebraically closed field of characteristic p. Let (B, 7’) be a Bore1 subgroup and maximal torus, and let X(T) be the character group. Let R be the set of roots of (G, T), R+ the set of roots of (BopP, T), S the set of simple roots in R+, and X’(T) the set of dominant characters. Let 1 be a character on B which is an element of X+(T), and let Mn be the irreducible G-module of highest weight 1. W is the Weyl group of (G, T), wO is the longest word in W, and yl’ is the coroot of a root y. In [4], we constructed the family of operators {Jw ( MJE W) on the set X(T) Oz [w (see the Preliminary section). For 1 E X(T) in generic, dominant position, we showed that {MJw l}w.E ,+, is a set of (multiplicity one) composition factors of H”(J) (here, we call those factors the scaffolding factors of H”(l)), and that H”“‘(w. 1) has socle M,., and unique top factor M,wow.n. At the same time, that serves to determine the top and socle factors of the image of an intertwining between any two higher cohomology modules. In this paper, we augment that information about the image of the non-zero intertwining cp: H”“‘l’(w, .A) -+ H’(“‘z)( w2 . A) by determining which of the scaffolding factors appear in the image (Section 2). In 1.1, we show the following facts about extensions among the scaffolding pieces: if Ext,!JM,%, .1, MJw2.J is non-zero, then w2 = SOW,, for