Abstract

LET % ’ be a highest weight category [CPSl] with finitely many simple objects and such that every object has finite length. Then Ce = mod- S for a finite dimensional quasi-hereditary algebra S. In a series of papers, [CPS4,5,6,7], we have studied various Kazhdan-Lusztig theories for %. This concept arises naturally in, for example, the modular representation theory of semisimple algebraic groups. As shown in [CPS4; § 41, the existence of a Kazhdan-Lusztig theory in this case is equivalent to the validity of the Lusztig conjecture (and leads to a considerable simplification of that conjecture). Also, the theory permits a combinatorial calculation of the Yoneda extension groups for irreducible representations with regular high weight in the Jantzen region. More generally, as shown in [CPS7; 5 21, if (e has a Kazhdan-Lusztig theory, its homological dual is also a highest weight category.

Citations