I am reading Humphrey's book "Representations of semisimple Lie algebras in the BGG Category O" on Page 189, Proposition 9.6, where he remarked that "Note that if we had developed the full BGG resolution of M_I(\lambda), this would correspond to an exact sequence completing the one in Theorem 9.4(b)." I am interested in this idea because this would prove directly that $$ ch M_I(\lambda)=\sum_{w\in W_I}(-1)^{\ell(w)}ch M(w\cdot\lambda) $$ I wonder if it is clear that we do have a full BGG resolution of L_I(\lambda) or it is still an open question for some reason. Why can't we just get the full BGG resolution of L_I(\lambda) by applying Corollary 6.5 (Page 114) in that book to the Levi subalgebra? Did I miss some important points here?

I am reading Humphrey's book "Representations of semisimple Lie algebras in the BGG Category O" on Page 189, Proposition 9.6, where he remarked that "Note that if we had developed the full BGG resolution of $L_I(\lambda)$, this would correspond to an exact sequence completing the one in Theorem 9.4(b)." I am interested in this idea because this would prove directly that $$ ch M_I(\lambda)=\sum_{w\in W_I}(-1)^{\ell(w)}ch M(w\cdot\lambda) $$ I wonder if it is clear that we do have a full BGG resolution of $L_I(\lambda)$ or it is still an open question for some reason. Why can't we just get the full BGG resolution of $L_I(\lambda)$ by applying Corollary 6.5 (Page 114) in that book to the Levi subalgebra? Did I miss some important points here?