# Extensions of modules over universal enveloping algebra with fixed central action

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).

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 $$\chi$$.