Localization for quantum groups at a root of unity

Backelin, Erik¹ ; Kremnizer, Kobi²

¹ Departamento de Matemáticas, Universidad de Los Andes, Carrera 4, 26-51, Bogota, Colombia
² Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139-4307

Journal of the American Mathematical Society, Tome 21 (2008) no. 4, pp. 1001-1018

Résumé

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.

Détail

DOI : 10.1090/S0894-0347-08-00608-5

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