In the paper: Kostant modules in blocks of category $\mathcal{O}^\mathfrak{p}$

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 $\mu\in\mathfrak{h}^*$?


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