Let $\mathcal{O}^{\mathfrak{p}}$ be a parabolic subcategory (see, Humphrey's BGG's Ch.9) of the BGG category $\mathcal{O}$ (w.r.t the Cartan $\mathfrak{h}$ and Borel $\mathfrak{b}$) over a finite-dimensional,complex, semisimple Lie algebra $\mathfrak{g}$.

First, assume that $\mathfrak{p} = \mathfrak{b}$. It follows from Bott's theorem that for all $n\in \mathbb{N}$ \begin{align} Ext^n_{\mathcal{O}}(M(\mu),N) \cong H^n(\mathfrak{n},N)_{\mu}, \end{align} where $N\in \mathcal{O}$, and $M(\mu)$ is the Verma module, and $\mathfrak{n}$ is the nilradical, also see Humphreys's BGG's Section 6.15.

This can be viewed as a simplest version for Vogan's interpretation by setting $\mathfrak{l} = \mathfrak{h}$

\begin{align} Ext^n_{\mathcal{O}}(M(\mu),N) \cong Hom_{\mathfrak{l}}(L(\mathfrak{l}, \mu),H^n(\mathfrak{n},N)), \end{align} for all $n\in \mathbb{N}$.

$\textbf{Question: Is it true for general parabolic subalgebra?}$

Thanks!

Let $\mathcal{O}^{\mathfrak{p}}$ be a parabolic subcategory (see, Humphrey's BGG's Ch.9) of the BGG category $\mathcal{O}$ (w.r.t the Cartan $\mathfrak{h}$ and Borel $\mathfrak{b}$) over a finite-dimensional,complex, semisimple Lie algebra $\mathfrak{g}$.

First, assume that $\mathfrak{p} = \mathfrak{b}$. It follows from Bott's theorem that for all $n\in \mathbb{N}$ \begin{align} Ext^n_{\mathcal{O}}(M(\mu),N) \cong H^n(\mathfrak{n},N)_{\mu}, \end{align} where $N\in \mathcal{O}$, and $M(\mu)$ is the Verma module, and $\mathfrak{n}$ is the nilradical, also see Humphreys's BGG's Section 6.15.

Namely, this is a special case in the following equality by letting $\mathfrak{l} = \mathfrak{h}$

\begin{align} Ext^n_{\mathcal{O}}(M(\mu),N) \cong Hom_{\mathfrak{l}}(L(\mathfrak{l}, \mu),H^n(\mathfrak{n},N)), \end{align} for all $n\in \mathbb{N}$.

$\textbf{Question: Is it true for general parabolic subalgebra?}$

Thanks!