Let $\mathfrak{g}$ be a Lie algebra over $\mathbb{C}$, $\mathfrak{z}$ be the center of $\text{U}(\mathfrak{g})$, and $M_1$, $M_2$ be $\text{U}(\mathfrak{g})$-modules on which $\mathfrak{z}$ acts by a fixed character, say, $\chi$. I'm wondering if there are some theories (and references) about $\mathrm{Ext}^1_{\chi}(M_1,M_2)$, the subgroup of $\mathrm{Ext}^1_{\text{U}(\mathfrak{g})}(M_1,M_2)$ consisting of those on which $\mathfrak{z}$ acts by $\chi$? For example, (vaguely) can it be calculated by certain kinds of relative Lie algebra cohomology? If it is helpful, we can assume $\mathfrak{g}$ bo be semi-simple, and $M_i$ to be objects in the category $\mathcal{O}$ (but the extension groups are taken in the category of $\text{U}(\mathfrak{g})$-modules). Thanks!

Let $\mathfrak{g}$ be a Lie algebra over $\mathbb{C}$, $\mathfrak{z}$ be the center of $\text{U}(\mathfrak{g})$, and $M_1$, $M_2$ be $\text{U}(\mathfrak{g})$-modules on which $\mathfrak{z}$ acts by a fixed character, say, $\chi$. I'm wondering if there are some theories (and references) about $\mathrm{Ext}^1_{\chi}(M_1,M_2)$, the subgroup of $\mathrm{Ext}^1_{\text{U}(\mathfrak{g})}(M_1,M_2)$ consisting of those on which $\mathfrak{z}$ acts by $\chi$? For example, (vaguely) can it be calculated by certain kinds of relative Lie algebra cohomology? If it is helpful, we can assume $\mathfrak{g}$ to be semisimple, and $M_i$ to be objects in the category $\mathcal{O}$ (but the extension groups are taken in the category of $\text{U}(\mathfrak{g})$-modules).