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.