# About Extension group and weights in $\mathcal{O}^\mathfrak{p}$

Denote $$M_I(\lambda)$$ be the generalized Verma module with highest weight $$\lambda$$ and $$L(\mu)$$ is the simple highest weight module with highest weight $$\mu$$. Suppose $$\text{Ext}_{\mathcal{O}^\mathfrak{p}}^i\left(M_I(\lambda),L(\mu)\right)\neq 0$$ for some $$i$$, does this implies $$\lambda\le \mu$$ or $$\lambda=w\cdot\mu$$ for some $$w\in W$$?

See proof of Theorem 6.11 of Representations of semisimple Lie algebras in the BGG category $$\mathcal{O}$$ by James E. Humphreys. This theorem proves what you want in the case $$\mathfrak{p}$$ is a Borel subalgebra ($$\mathfrak{p} = \mathfrak{b}$$). You just need existence of projective covers in parabolic setting, statement about $$\mathrm{Ext}^1$$ (I think both can be found in the same book) and long exact sequence for $$\mathrm{Ext}_{\mathcal{O}_\mathfrak{p}}$$ (which is standard homological algebra?).