Ringel duality and Kazhdan–Lusztig theory

Abstract

Ringel duality exhibits a symmetry for quasi-hereditary algebras, which, in particular, is of interest for blocks of the BGG-category O and for Schur algebras of classical groups. This symmetry is used to phrase (Kazhdan-)Lusztig conjecture in terms of maps between tilting modules and also in terms of composition factors occuring in certain layers of good or cogood filtrations of tilting modules. The conditions make sense for centralizer subalgebras as well where they can be formulated in terms of non-existence of certain uniserial submodules. Hence, the validity of (Kazhdan-)Lusztig conjecture is equivalent to a 'regularity' condition on the structure of tilting modules.