See section 9.7. (Relative Kazhdan-Lusztig Theory) of his book [1], and the references therein.
[1] J. Humphreys. Representations of semisimple Lie algebras in the BGG category ORepresentations of semisimple Lie algebras in the BGG category $\mathscr O$. Graduate Studies in Mathematics, 94. American Mathematical Society, Providence, RI, 2008.