Extensions of modules over universal enveloping algebra with fixed central action

Question

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).

Answer 1

The question is indeed somewhat too vague, since a knowledge of certain Ext$^1_\chi$ would be enough to prove the Kazhdan-Lusztig Conjecture: take $M_1$ to be a Verma module and $M_2$ to be a simple highest weight module on which U($\mathfrak{g}$) acts with the the same central character $\chi$.

References include papers by David Vogan in the 1970s which approached the conjecure. See also my graduate textbook GSM 94 (Amer. Math. Soc., 2008), Chap. 6 and 8, citing early work by P. Delorme in 6.14 and by D. Vogan in 8.10-8.11, plus lists of errata on the AMS page and my homepage.