The category $\mathcal{O}$ is the category of all finitely generated, locally $\mathfrak{b}$-finite and $\mathfrak{h}$-semisimple $\mathfrak{g}$-modules, where $\mathfrak{g}$ is a complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$ and Borel subalgebra $\mathfrak{b}$ containing $\mathfrak{h}$.

Let $M(\lambda)$ be the Verma module with highest weight $\lambda$, $L(\lambda)$ be its unique simple quotient.

Does $\text{Ext}^i_{\mathcal{O}}(M(\lambda),M(\mu))=0$ imply that $\text{Ext}^i_{\mathcal{O}}(M(\lambda),L(\mu))=0$?

  • $\begingroup$ I don't know whether counterexamples exist, butTheorem 6.11 in my 2008 GSM text places some restrictions on what these would look like. See also the series of papers on homology of $\mathcal{O}$ by V. Mazorchuk et al. – $\endgroup$ – Jim Humphreys Mar 12 at 20:16

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