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.
