Maximal Submodules and the Second Loewy Layer of Standard Modules

The paper begins in §1 with a foundational discussion of a new notion, that of a semistandard filtration in a highest weight category. The main result is Theorem 1, which says that “multiplicities” of standard modules in such filtration are well-defined. In §2, we specialize to the case of semistandard filtrations of maximal submodules of standard modules. The main result is Theorem 2, which under a simple Kazhdan-Lusztig theory hypothesis, characterizes standard module multiplicities in terms of Ext between irreducible modules, or equivalently, second Loewy layers of standard modules. This theorem was really the starting point of this paper. The author found it while trying to explain the mysterious frequency of “0 or 1” answers for semistandard module multiplicities in maximal submodules of standard modules. The next section, §3, applies the theory of §2 to obtain some general inequalities on the behavior of Ext in the presence of a suitable exact functor. This theory is used in §4 to attack some well-known issues about Ext and parity conditions involving standard and irreducible modules with singular high weights in the presence of the Lusztig conjecture for representations of semisimple algebraic groups in positive characteristic. Some indirect evidence is given for an expected parity behavior for weights on a wall, and this behavior is formalized as Conjecture 1. We also offer a suggested possible direction for its future proof. The usefulness of such parity behavior is exhibited by Theorem 3, which shows Conjecture 1 implies some new results relating Ext been standard and irreducible modules in the singular weight cases to Ext for related modules with nonsingular high weights. Proposition 8 proves a special case of the theorem, without the use of Conjecture 1.