Twin categories in representation of Lie algebra

Let $\mathfrak{g}=\mathfrak{n}^-\oplus\mathfrak{h}\oplus\mathfrak{n^+}$ be a triangular decomposition of semisimple Lie algebra. Let $\mathcal{Z}$ be the central of universal envoloping Lie algebra of $\mathfrak{g}$.

Let $\mathcal{C}$ be the category of representations of $\mathfrak{g}$, on which $\mathfrak{n ^+}$ and $\mathfrak{h}$ acts locally finite, and $\mathcal{Z}$ acts semisimplely.

Let $\mathcal{D}$ be the another category of representations of $\mathfrak{g}$, on which $\mathfrak{n ^+}$ and $\mathcal{Z}$ acts also locally finitely, and $\mathfrak{h}$ acts semisimply. If I don't make mistake, $\mathcal{D}$ should be called category $\mathcal{O}$.

Claim. $\mathcal{C}$ is equivalent to $\mathcal{D}$.

Do I formulate the problem correctly? About the proof of this theorem, where is it written?

• What field? Should "Let D" just be "D" in the claim? Commented Apr 9, 2010 at 18:41
• Over complex field
– JJH
Commented Apr 9, 2010 at 19:08
• Probably you want to require your modules to be finitely generated. In any case, it's a good idea to look at a short paper by Soergel in which he notes both positive and negative results closely related to the question you raise: MR0872544 (88c:17011) Soergel, Wolfgang, Équivalences de certaines catégories de ${\germ g}$-modules. (French. English summary) [Equivalences of certain categories of $\mathfrak{g}$-modules], C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), no. 15, 725--728. Commented Apr 10, 2010 at 19:37
• Dear Professor Humphreys, this is the proof I expected. I had heard of the idea roughly, but didn't know where it was written. It's nice to know another proof by D-module, which is apparently more elegant.
– JJH
Commented Apr 12, 2010 at 13:45

The equivalence (or something very close, I haven't checked carefully what is written) follows from Beilinson-Bernstein localization. The two categories can be realized roughly speaking as D-modules on B\G/N and N\G/B, and the equivalence comes from the interchange of the two sides. Slightly more precisely, Beilinson-Bernstein tells us that (assuming we ignore singular infinitesimal characters, where things need to be slightly modified) if we want representations on which Z acts semisimply, we look at twisted D-modules on G/B with twisting given by (a lift from h^/W to h^ of) the eigenvalues of the Z action. Equivalently these are D-modules on G/N which are weakly H-equivariant - meaning locally constant along the fibers G/N-->G/B, and h acts with strictly prescribed semisimple monodromy-- ie we presecribe monodromy along these fibers. If we want representations with locally finite Z action, we really just look at D-modules on G/N and ask for them to be locally constant along the fibers but don't strictly presecribe monodromies. Now the two conditions you give for representations correspond to asking for these D-modules to be N-equivariant in the strict case or B-equivariant in the locally finite case. In any case the whole picture is symmetric under exchanging left and right, hence the equivalence.

(There are two other categories which are symmetric under exchange of left and right -- if we impose the weak/locally finite conditions on both sides we get the category of Harish-Chandra bimodules -- ie (g+g,G)-Harish Chandra modules -- which correspond to representations of G considered as a real Lie group. If we impose strict conditions on both sides we get the Hecke category, which appears as intertwining functors acting on categories of representations and is the subject of Kazhdan-Lusztig theory. Category O, in these two forms, is some kind of intermediate form -- both of the above are monoidal categories and Category O is a bimodule for them, with your involution exchanging the two actions..)

As for a reference, this is standard but I don't know the proper reference. Similar things are discussed in the Beilinson-Ginzburg-Soergel JAMS paper on Koszul duality patterns in representation theory or the Beilinson-Ginzburg paper on wall-crossing functors (available on the arxiv). I presume Ben Webster will let us know..

• Thanks, it's very informative. But I need some time to understand. Here are some questions: 1. How to relate D-modules on G/N with monodromy locally finite along fibers, with representation of Lie algebra? Just by taking global sections functor? I don't whether we still have localization theorem. 2. I roughly know, to relate two types of D-modules by taking involution. But I can't see the picture, how it interchange semisimple/locallly finite condition?
– JJH
Commented Apr 9, 2010 at 21:20
• About the end of your second paragraph, how do you relate category O to category of Harish-chandra bimodule? You have mentioned many categories, I simply don't know which two monoidal categories you mean. If I understan correctly, on my question 2, that's trivial. Actually what you say is that, for special case, on G, by involution we will get $B\times (B,\lambda)$ -equivariant D-module from $(B,\lambda)\times B$ -equivariant D-module, where $\lambda$ is some dominant weight of B. I'm a little confused, given a infinitesimal character, it seems we can associate many different TDO on G/B?
– JJH
Commented Apr 9, 2010 at 22:02
• 1. The functor is still global sections.. the localization theorem holds for any regular generalized infinitesimal character, not just strict/semisimple ones.. 2. I don't know how to think about it algebraically, but on one side you have strict H equivariance, on the other only a weak form, so for one you get semisimplicitly (ie the representation integrates to the torus group) and the other local finiteness only. Commented Apr 9, 2010 at 22:19
• Harish Chandra bimodules are D-modules on N\G/N which are weakly H equivariant on both sides --- or if you prefer, D-modules on G(G/N x G/N) with weak equivariance. This is a monoidal category by convolution, and acts on D-modules on N\G/K for any K (eg B) which are similarly H-monodromic. You're right of course, to localize on flag manifolds we need to lift infinitesimal characters to a choice of TDO (which is generically a W-ambiguity) Commented Apr 9, 2010 at 22:22
• Yes weakly H-equivariant=H-monodromic, and Hecke category is the derived category of D-modules on B\G/B (probably everything I say should be derived, or you need to be a little more careful with monoidal structures). Any category of D-modules on K\G/B eg is a Hecke module category, so eg category O. The action is not by tensor product, but by convolution -- categorification of the classical notion of Hecke or intertwining operators. It's monoidal but not symmetric (its K-group is the Weyl group) so Tannakian formalism doesn't apply. Commented Apr 10, 2010 at 0:02

Following David's suggestion, I'll point out that this is a theorem of Soergel. That paper is in French, but an English account of a related (and more general) construction is given in the paper of Soergel and Milicic. While David's explanation is quite nice, the references below show this isn't an intrinsically geometrical result; Soergel gives completely algebraic proofs.

• The added reference is helpful. It's a somewhat amended version of the paper published in Comm. Math. Helv. 72 (1997), 503-520 and appears on Soergel's homepage: home.mathematik.uni-freiburg.de/soergel Commented Apr 13, 2010 at 21:09