Ringel duality and Kazhdan-Lusztig theory. (Q701231)

Let \(A\) be a quasi-hereditary algebra with respect to some partial order \(<\) on the set \(\Lambda\) of iso-classes of simple modules. Assume that \(A\) has a duality and that for all \(\lambda,\mu\in\Lambda\) such that \(\lambda<\mu\) and \(\lambda\) and \(\mu\) are neighbors with respect to \(<\) the composition multiplicity of the simple module \(L(\lambda)\) in the standard module \(\Delta(\mu)\) equals \(1\). Let further \(B\) be the Ringel dual of \(A\). It is known that for the blocks of the BGG category \(\mathcal O\) the following condition is equivalent to the Kazhdan-Lusztig Theorem: (1) for all \(\lambda,\mu\in\Lambda\) such that \(\lambda<\mu\) and \(\lambda\) and \(\mu\) are neighbors with respect to \(<\) the \(B\)-cohomology space \(\text{Ext}_B^1(L(\mu),L(\lambda))\) is not zero. If \(B\) is a Schur algebra then (1) is equivalent to the Lusztig conjecture. In the paper under review the author presents five equivalent reformulations of the condition (1) in terms of certain structural properties of \(A\), \(B\), and tilting \(A\)-modules.