# Isomorphisms between extension group and $\mathfrak{u}$-cohomology

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!

• I never got far enough into this literature to give you an expert answer, but a basic source is Kostant's fundamental 1961 Annals paper: ams.org/mathscinet-getitem?mr=0142696 Even though some of his notation is now obsolete, he influenced a great many people who wrote related papers afterward. The key point, I think, is that he studied cohomology for nilradicals of parabolic subalgebras in a way that explained Bott's observations when the parabolic is a Borel subalgebra. Later work of Knapp and Wallach might be most helpful in rewriting Kostant's ideas. – Jim Humphreys May 18 '16 at 22:29