# Extension of Verma modules

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

• 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. – – Jim Humphreys Mar 12 '19 at 20:16