So, in learning about category $\mathcal{O}$ representations of a semisimple Lie algebra $\mathfrak{g}$, I've come across two natural kinds of subcategories, and I think I'm confused about their structure and the relationship between them. (In particular, if they are equivalent or not).


Fix $\lambda \in \mathfrak{h}^{\*}$ and let $\chi\_{\lambda}:Z(\mathfrak{g}) \rightarrow \mathbb{C}$ denote the corresponding central character ($Z(\mathfrak{g})$ denoting the center of $U\mathfrak{g}$)


**Type 1:** Modules $M$ such that $(z - \chi\_{\lambda}(z))\cdot m = 0$ for all $m \in M$, $z \in Z(\mathfrak{g})$. Let's call this $\text{Mod}\_{\mathcal{O}}(\mathfrak{g},\lambda)$


**Type 2:** Modules $M$ such that $(z - \chi\_{\lambda}(z))^{n} \cdot m = 0$ for all $m \in M$, $z \in Z(\mathfrak{g})$, where $n$ is an integer possibly depending on $z$ and $m$. Let's call this $\mathcal{O}\_{\lambda}$.


Now one clear reason to prefer the $\mathcal{O}\_{\lambda}$ is that category $\mathcal{O}$ is a direct sum of these subcategories. Additionally, if $L \in \mathcal{O}\_{\lambda}$ is simple, it's projective cover will also be in $\mathcal{O}\_{\lambda}$, but not necessarily in $\text{Mod}\_{\mathcal{O}}(\mathfrak{g},\lambda)$.


But it seems like, in some cases, these categories should be equivalent?


For simplicity, let $\lambda = 0$. If I'm understanding correctly:


**1.** Beilinson-Bernstein says that there is an equivalence of categories: $\text{Mod}\_{\mathcal{O}}(\mathfrak{g},0) \cong \text{Mod}\_{B}(D\_{X})$, where the second thing is weakly $B$-equivariant $D$-modules on the flag variety $X$.


**2.** Riemann-Hilbert says that there is an equivalence $\text{Mod}\_{B}(D\_X) \cong \mathcal{P}\_{B}(X)$, the second category being perverse sheaves on $X$ whose cohomology is constant along $B$-orbits.


**3.** In the BGS paper on Koszul duality, it is claimed that $D^{b}(\mathcal{O}\_{0}) \cong D^{b}\_{B}(X)$, the latter being the bounded derived category of constructible sheaves on $X$ whose cohomology is constant along $B$-orbits, and this congruence respects the $t-$structures giving $\mathcal{O}\_{0} \cong \mathcal{P}\_{B}(X)$.


Is this all correct? If it is, it would seem to establish an equivalence $\mathcal{O}\_{0} \cong \text{Mod}\_{\mathcal{O}}(\mathfrak{g},0)$. Which seems plausible, but also somewhat strange, since one is a strict subcategory of the other (not that that by itself would imply non-equivalence..). And even trying the simplest case, $\mathfrak{sl}\_2$, I am unable to realize this equivalence, for example, the category $\text{Mod}\_{\mathcal{O}}(\mathfrak{sl}\_2,0)$ only seems to have four indecomposables (since it doesn't have the projective cover of $L(-2)$).



Confusion about Subcategories of Category $\mathcal{O}$
The source of your confusion is the difference between B-equivariance and being smooth along B-orbits. Since in step 3, you want only smoothness along B-orbits, you have to do the same thing in step 1. In order to do that, you need to take some modules not in category $\mathcal{O}$, and instead consider modules where the Cartan subalgebra $\mathfrak{h}$ acts locally finitely (not necessarily semi-simply!), and the subalgebra $\mathfrak{n}$ nilpotently. 


The important point here is that now there's no containment of Type 1 or Type 2 modules: there are some modules on which the center acts semi-simply and the Cartan subalgebra does not, and some where it's *vice versa*. These categories are equivalent by an old [theorem of Soergel](http://www.ams.org/mathscinet-getitem?mr=872544). This theorem is secretly hiding in the result cited in BGS.


