Abstract
We study the category \(\mathcal I _{\mathrm{gr }}\) of graded representations with finite-dimensional graded pieces for the current algebra \(\mathfrak{g }\otimes \mathbf{C }[t]\) where \(\mathfrak{g }\) is a simple Lie algebra. This category has many similarities with the category \(\mathcal O \) of modules for \(\mathfrak{g }\), and in this paper, we prove an analog of the famous BGG duality in the case of \(\mathfrak{sl }_{n+1}\).
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Acknowledgments
We thank Boris Feigin for stimulating discussions. It is a pleasure for the second, third and fifth authors to thank the organizers of the trimester “On the interactions of Representation theory with Geometry and Combinatorics,”at the Hausdorff Institute, Bonn, 2011, when much of this work was done. The fourth and fifth authors also thank Giovanni Felder for his support and for the hospitality of the ETH Zurich.
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A.B. was partially supported by DMS-0800247 and DMS-1101507.
V.C. was partially supported by DMS-0901253.
S.L. was partially supported by RFBR-CNRS-11-01-93105 and RFBR-12-01-00944.
A.K was supported by the grants NSh-3349.2012.2, RFBR-10-01-00836, RFBR-CNRS-10-01-93111, RFBR-CNRS-10-01-93113, and by the Simons Foundation.
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Bennett, M., Berenstein, A., Chari, V. et al. Macdonald polynomials and BGG reciprocity for current algebras. Sel. Math. New Ser. 20, 585–607 (2014). https://doi.org/10.1007/s00029-013-0141-7
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DOI: https://doi.org/10.1007/s00029-013-0141-7