# About isomorphism of Extension groups between Category $\mathcal{O}^\mathfrak{p}$ and $\mathcal{O}$

In section 8.2 (p.22), I use the notation in Humphrey's Category $$\mathcal{O}$$ book.

Then the passage said the following: The general formula follows from the following isomorphism (which is proved using a Lyndon-Hochschild-Serre spectral sequence argument as in [ES1, Chap. 15])

$$\text{Ext}^i _{\mathcal{O}^\mathfrak{p}}(M_I(w_Iy\cdot\mu), L(w_Iw\cdot\mu)) \cong\text{Ext}^i _{\mathcal{O}}(M(w_Iy\cdot\mu), L(w_Iw\cdot\mu)),$$

where $$M(w_Iy\cdot\mu)$$ is the ordinary Verma module of highest weight $$w_Iy\cdot\mu$$ in the block $$\mathcal{O}_\mu$$ of ordinary category $$\mathcal{O}$$."

Note that $$M_I(w_Iy\cdot\mu)$$ is the parabolic Verma module defined in Humphrey's book Ch. 9. And $$L(\lambda)$$ is the unique simple subquotient of both $$M_I(\lambda)$$ and $$M(\lambda)$$.

Does anyone know how to use Lyndon-Hochschild-Serre spectral sequence argument to show the isomorphism for any $$y,w\in {}^SW^J$$ and any anti-dominant integral weight $$\mu\in\mathfrak{h}^*$$?