ABSTRACT KAZHDAN-LUSZTIG THEORIES

Abstract

In this paper, we prove two main results. The first establishes that Lusztig's conjecture for the characters of the irreducible representations of a semisimple algebraic group in positive characteristic is equivalent to a simple assertion that certain pairs of irreducible modules have non-split extensions. The pairs of irreducible modules in question are those with regular dominant weights which are mirror images of each other in adjacent alcoves (in the Jantzen region). Secondly, we establish that the validity of the Lusztig conjecture yields a complete calculation of all Yoneda Ext groups between irreducible modules having regular dominant weights in the Jantzen region. These results arise from a general theory involving so-called Kazhdan-Lusztig theories in an abstract highest weight category. Accordingly, our results are applicable to a number of other situations, including the Bernstein-Gelfand-Gelfand category for a complex Lie algebra and the category of modules for a quantum group at a root of unity.