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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 $w\in W$?

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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?).

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