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 $L(-2)$).

  • 2
    $\begingroup$ Sorry- 'Koszul Duality Patterns in Representation Theory', by Beilinson, Ginzburg and Soergel (Journal of the American Mathematical Society, volume 9, number 2, April 1996). $\endgroup$ – Alex Zorn Oct 6 '15 at 5:41
  • $\begingroup$ But the fact that one has projective covers of simples and the other does not should rule out equivalence, should it not? $\endgroup$ – Tobias Kildetoft Oct 6 '15 at 7:10
  • $\begingroup$ @Alex: I've rewritten my attempted answer and start by pointing out that your header is itself misleading. Sorry for the belated remarks. $\endgroup$ – Jim Humphreys Oct 19 '15 at 20:12

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. This theorem is secretly hiding in the result cited in BGS.

  • $\begingroup$ Thanks for the answer Ben, this clears a lot up. I definitely admit to being confused at the distinction between B-equivariance and smooth along B-orbits- I think I sort of swept it under the rug and assumed it was all the same. $\endgroup$ – Alex Zorn Oct 10 '15 at 17:25

EDIT: To compensate for my attempted answer, which mainly added further confusion, I'll substitute the following remarks.

Note especially that on the algebraic side the confusion starts in the wording of your header. Following Ben's helpful reminder about the 1986 Comptes Rendus note by Soergel (which I then recalled I had written a review of at the time), I'd emphasize that Soergel's category equivalence is more subtle than your set-up suggests.

Here you have a semisimple Lie algebra $\mathfrak{g}$ with Cartan subalgebra $\mathfrak{h}$, where your Type 2 subcategory $\mathcal{O}_\lambda$ (for $\lambda \in \mathfrak{h}^*$) is a "block" of the BGG category $\mathcal{O}$ (but using the Moscow/Paris convention which incorporates a $\rho$ shift so that the weight $\lambda =0$ becomes regular). However, your Type 1 category of $\mathfrak{g}$-modules is incompletely defined: here $\mathfrak{h}$ need not act semisimply and the category usually isn't a subcategory of $\mathcal{O}$.

What Soergel does to get a subtle category equivalence for regular $\lambda$ (not valid in general as seen when $\lambda = -\rho$)) is to embed both categories of $\mathfrak{g}$-modules in a larger category of Harish-Chandra bimodules and then relate them indirectly.

One other clarification is that Beilinson-Bernstein didn't explicitly deal with an entire module category or its blocks in their localization, since their main emphasis was on simple modules and Verma modules, especially in the principal block $\mathcal{O}_0$ and then more generally for $\mathcal{O}_\lambda$ in terms of twisted differential operators. But projectives and such don't seem to play a direct role here, as they do in $\mathcal{O}$.

  • $\begingroup$ Thanks for the answer. My confusion stemmed from the fact that the map $U(\mathfrak{g}) \rightarrow D_X$ has $z - \chi_0(z)$ in its kernel for all $z \in Z$, which would indicate my 'Type 1' category described above. However, a more careful reading of the Hotta et. al. book shows that they don't look at category O, they look at ALL modules where the center acts semisimply by a fixed character. $\endgroup$ – Alex Zorn Oct 10 '15 at 17:23

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