$\newcommand{\g}{\mathfrak{g}}$Setting: $\mathfrak{g}$ is a semisimple complex Lie algebra. Here $\chi_\lambda$ denotes the central character corresponding to the action of $Z(\g)$ on a highest weight module of weight $\lambda$, i.e. Harish-Chandra says $\chi_\lambda=\chi_\mu$ if and only if $\lambda$ and $\mu$ are related by a SHIFTED action of the Weyl group, i.e. $w(\lambda+\rho)-\rho=\mu$. I emphasize this because lots of people use another convention such that Harish-Chandra does not use the shifted action. This notational/conventional difference is the source of my question. I let $\operatorname{Mod}U(\g)/\chi$ denote the category of modules on which $Z(\g)$ acts by $\chi$, i.e. mod out by $z-\chi(z)$.

**Question:** Is the following true: that for $\lambda$ antidominant and regular (both in the unshifted sense), one has $\operatorname{Mod} U(\g)/\chi_{\lambda+\rho}\cong \operatorname{Mod} U(\g)/\chi_{\lambda-\rho}$?

The reason I ask this is because different sources state Beilinson–Bernstein localization using different conventions. I sat down and picked two sources (in case it matters, Yi Sun’s expository paper $\mathcal D$-Modules and Representation Theory and Miličić’s preprint Localization and Representation Theory of Reductive Lie Groups) and unwound the conventions and got that this must be true in order for both of them to be correct.

I think something like this is true for category $\cal O$ due to translation functors, but I’m not sure what I can say for all $U(\g)$ modules.

(I guess it’s also possible that this is false because I unwound the conventions incorrectly or because one of the sources I picked made a mistake in their conventions.)