Representations of Semisimple Lie Algebras in the BGG Category O

Discusses the Kazhdan-Lusztig Conjecture on characters of simple highest weight modules for a semisimple Lie algebra $\mathfrak{g}$ over $\mathbb {C}$. This work covers topics including parabolic category $\mathscr {O}$, projective functors, tilting modules, twisting and completion functors and Koszul duality theorem of Beilinson-Ginzburg-Soergel.