Let $\mathfrak{g}$ be a solvable lie algebra and $\lambda\in (\mathfrak{g}/[\mathfrak{g},\mathfrak{g}])^*$ be the character of $\mathfrak{g}$. How to compute homology for $\mathbb{C}_\lambda$, the simple $\mathfrak{g}-$module with character $\lambda$?
I have faced strange behavior of homology, while calculating them manually for 2 and 3-dimensional algebras and cannot solve it in full generality. Thanks in advance!