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 $L(-2)$).