Representations of Semisimple Lie Algebras in the BGG Category O

antidominant weight arbitrary BGG Reciprocity BGG resolution block Ox Bruhat ordering central character Chapter cohomology composition factor multiplicities compute contravariant form coset D(ws defined denote dim Homo dimension direct sum dominant duality element embedding equivalent example Exercise Exto finite dimensional module fixed flag variety follows formal characters Grothendieck group highest weight module implies indecomposable induction injective integral weights involving isomorphic Jantzen filtration Kac-Moody algebras Kazhdan-Lusztig KL Conjecture KL polynomials Lemma Lie algebra Lie group linkage class Loewy maximal vector notation parabolic Verma modules positive roots projective functors projective module proof Proposition regular root system scalar self-dual short exact sequence shows simple modules simple roots simple submodule socle standard filtration subalgebra subcategory subgroup submodule summand tilting module translation functors U(g)-module Verma modules w₁ wall-crossing functors weight spaces Weyl group