Abstract
We establish a correspondence (or duality) between the characters and the crystal bases of finite-dimensional representations of quantum groups associated to Langlands dual semi-simple Lie algebras. This duality may also be stated purely in terms of semi-simple Lie algebras. To explain this duality, we introduce an “interpolating quantum group” depending on two parameters which interpolates between a quantum group and its Langlands dual. We construct examples of its representations, depending on two parameters, which interpolate between representations of two Langlands dual quantum groups.
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Acknowledgments
This work was begun while we were taking part in the Program on Combinatorial Representation Theory held at MSRI in the Spring of 2008. We thank the organizers of this Program for their invitations and MSRI for hospitality.
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E. Frenkel was supported in part by DARPA and AFOSR through the grant FA9550-07-1-0543 and by Fondation Sciences mathématiques de Paris.
D. Hernandez was supported partially by ANR through Project “Géométrie et Structures Algébriques Quantiques”.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Frenkel, E., Hernandez, D. Langlands duality for representations of quantum groups. Math. Ann. 349, 705–746 (2011). https://doi.org/10.1007/s00208-010-0541-3
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DOI: https://doi.org/10.1007/s00208-010-0541-3