ABSTRACT KAZHDAN-LUSZTIG THEORIES

In this paper, we prove two main results. The first establishes that L usztig's conjecture for the characters of the irreducible representati ons of a semisimple algebraic group in positive characteristic is equi valent to a simple assertion that certain pairs of irreducible modules have non-split extensions. The pairs of irreducible modules in questi on are those with regular dominant weights which are mirror images of each other in adjacent alcoves (in the Jantzen region). Secondly, we e stablish that the validity of the Lusztig conjecture yields a complete calculation of all Yoneda Ext groups between irreducible modules havi ng 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 app licable to a number of other situations, including the Bernstein-Gelfa nd-Gelfand category for a complex Lie algebra and the category of modu les for a quantum group at a root of unity.