## Localization for quantum groups at a root of unity

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- by Erik Backelin and Kobi Kremnizer
- J. Amer. Math. Soc.
**21**(2008), 1001-1018 - DOI: https://doi.org/10.1090/S0894-0347-08-00608-5
- Published electronically: June 19, 2008
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## Abstract:

In the paper *Quantum flag varieties, equivariant quantum $\mathcal {D}$-modules, and localization of Quantum groups*, Backelin and Kremnizer defined categories of equivariant quantum $\mathcal {O}_q$-modules and $\mathcal {D}_q$-modules on the quantum flag variety of $G$. We proved that the Beilinson-Bernstein localization theorem holds at a generic $q$. Here we prove that a derived version of this theorem holds at the root of unity case. Namely, the global section functor gives a derived equivalence between categories of $U_q$-modules and $\mathcal {D}_q$-modules on the quantum flag variety.

For this we first prove that $\mathcal {D}_q$ is an Azumaya algebra over a dense subset of the cotangent bundle $T^\star X$ of the classical (char $0$) flag variety $X$. This way we get a derived equivalence between representations of $U_q$ and certain $\mathcal {O}_{T^\star X}$-modules.

In the paper *Localization for a semi-simple Lie algebra in prime characteristic*, by Bezrukavnikov, Mirkovic, and Rumynin, similar results were obtained for a Lie algebra $\mathfrak {g}_p$ in char $p$. Hence, representations of $\mathfrak {g}_p$ and of $U_q$ (when $q$ is a $p$’th root of unity) are related via the cotangent bundles $T^\star X$ in char $0$ and in char $p$, respectively.

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## Bibliographic Information

**Erik Backelin**- Affiliation: Departamento de Matemáticas, Universidad de Los Andes, Carrera 4, 26-51, Bogota, Colombia
- Email: erbackel@uniandes.edu.co
**Kobi Kremnizer**- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139-4307
- Email: kremnize@math.mit.edu
- Received by editor(s): November 1, 2006
- Published electronically: June 19, 2008
- Additional Notes: The second author was supported in part by NSF grant DMS-0602007
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**21**(2008), 1001-1018 - MSC (2000): Primary 14A22, 17B37, 58B32; Secondary 20G42
- DOI: https://doi.org/10.1090/S0894-0347-08-00608-5
- MathSciNet review: 2425178