Let $\mathfrak{g}$ be a solvable Lie algebra over $\mathbb{C}$ and $\lambda\in(\mathfrak{g}/[\mathfrak{g},\mathfrak{g}])^*$ be the character of $\mathfrak{g}$. I'm interested in calculating homology for $\mathbb{C}_\lambda$, the one-dimensional $\mathfrak{g}-$module with character $\lambda$.

I have calculated homology manually for 2-dimensional algebra $\mathfrak{g}=\langle x,y \rangle$ with relation $[x,y]=y$. The thing surprised me is that homology is nontrivial only for $\lambda(x)=0$ or 1. The hypothesis is that homology is nontrivial, iff $\lambda$ is an eigenvalue of adjoint representation. I can't prove it (or find counterexample). I've tried to find the answer in books, but there is a lack on literature in homology theory for solvable lie algebras.