# Is there an analogue Beilinson-Bernstein localization for quantized enveloping algebra

I am completely a beginner in this field. I wonder know whether there is appropriate notion for quantum flag variety of finite dimensional Lie algebra. If so, what is the correspondent notion for "quantum differential operator on this quantum flag variety". If such notions exist. I think there should be some kind of quantum analogue of Beilinson-Bernstein localization for quantum group (or quantized enveloping algebra)?

The second question is related to the question D-module theory Scott Carnahan mentioned that the category of D-module can be taken as category of quasi coherent sheaves on DeRham stack. So, if there exits "quantum D-module". Then in this case, the category of D-module can be taken as category of quasi coherent sheaves on "quantized" DeRham stack? How do we define the quantized DeRham stack? Or more generally, is there an appropriate notion for "quantized" stack?

Yes, there exists the result for quantized enveloping algebra. It is developed by Lunts-Rosenberg and proved by them and Tanisaki.

There are several notions:

1. quantized flag variety
2. quantum D-module.

In the framework of noncommutative algebraic geometry. quantized flag variety is defined as a noncommutative projective scheme. It is a proj-category of quantized enveloping algebra of Lie algebra g. It is a noncommutative separated scheme with affine covers discovered by A.Joseph.

Lunts and Rosenberg defined differential calculus in noncommutative algebraic geometry. They introduced the noncommutative version of Grothendieck differential operators in differential operator on noncommutative ring and then applied this construction to define quantized D-modules in localization for quantum group. In this paper, they formulated the quantized Beilinson Bernstein localization for quantized enveloping algebra in generic case. They proved the global section functor is exact and conjectured it is indeed the correct quantized version, which means it is an equivalent.

Later, under this framework. Tanisaki proved in his paper The Beilinson-Bernstein correspondence for quantized enveloping algebras that the conjecture of Lunts-Rosenberg is indeed true. Moreover, Tanisaki proved this result in root of unity case,see D-modules on quantized flag manifolds at roots of 1.

More comments: In the paper of Lunts-Rosenberg, they pointed out the localization for the quantum group sl2 was constructed by "hand" by T.J.Hodges.

• The link to Tanisaki's paper at math.ecnu.edu.cn is broken. It seems it should point to Tanisaki's paper (notes?) in a 2009 workshop at East China Normal University. I believe these notes were published in Volume 2 of the series Surveys of Modern Mathematics, titled Lie Theory and Representation Theory. Chapter 4 is written by Tanisaki, and is titled $D$-modules and Representation Theory. May 24 at 23:55
• Tanisaki has also written a series of papers titled Differential operators on quantized flag manifolds at roots of unity: Part I (2012), Part II (2014), and Part III (2021). May 25 at 0:00

Yes, it's discussed in the paper

• Erik Backelin, Kobi Kremnizer, Quantum flag varieties, equivariant quantum D-modules and localization of quantum groups, Advances in Mathematics 203 Issue 2 (2006) pp 408–429, doi:10.1016/j.aim.2005.04.012, arXiv:math/0401108.

[Edit: removed an attribution after Shizhuo's correction]

Kremnizer gave a nice course where he worked through the examples of G=SL_2, G/B=CP^1 in complete detail. I have some incomplete notes if you want to email me (I'd rather not post them online since they are still being revised).

To answer your question briefly about what the notion of quantum differential operators are for the flag variety, here is a rough outline:

1. First, define quantum differential operators on G. This is done by constructing the so-called Heisenberg double (just another name in this specific situation for the semi-direct product, also called smash prodcut) D(U_q,O_q) of the quantum group U_q with its dual Hopf algebra O_q, where U_q acts on O_q by the left-regular action (X.f)(y):=f(S(X)y), here X is the antipode.

2. O_q(G) has a sub-algebra called O_q(B) which is a quantization of the functions on the Borel.

3. In sufficiently nice cases in algebraic geometry, one can identify D(X/G)-modules where X is some variety with G-action as D(X)-modules M, together with an O(G) co-action, and a compatibility condition. The case G/B is such a situation classically, so one defines D_q(G/B)-modules to be D_q(G)-modules with a O_q(B) co-action plus compatibility.

By the way, there are some papers of Varagnolo and Vasserot, notably https://arxiv.org/abs/math/0603744, which discuss D_q(G) and might introduce you to some tricks people use in the area.

• No, this result is due to Lunts-Rosenberg-Tanisaki.Lunts-Rosenberg developed the differential calculus in noncommutative algebratic geometry and defined quantized flag variety as noncommutative projective scheme. They formulated the quantized version of BB-localization and proved this functor is exact. Later, Tanisaki proved this functor is indeed an equivalence. Backelin and Kremnizter used another approach to prove the same results. Jan 11, 2010 at 23:32
• Shizhuo, thanks very much for the correction! I foresee a question being posted in the near future by me about the parallels between D_q(G) and the application of quantum differential calculus (about which I know quite little). Jan 12, 2010 at 15:47