Macdonald polynomials and BGG reciprocity for current algebras

We study the category Igr of graded representations with finite-dimensional graded pieces for the current algebra g ⊗ C[t] where g is a simple Lie algebra. This category has many similarities with the categoryO of modules for g, and in this paper, we prove an analog of the famous BGG duality in the case of sln+1. Mathematics Subject Classification 17B65 · 81R10 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. M. Bennett IMECC-UNICAMP Rua Sergio Buarque de Hollanda, 651 Barao Geraldo, Campinas, SP13083-859, Brazil e-mail: [email protected] V. Chari Department of Mathematics, University of California, Riverside, CA 92521, USA e-mail: [email protected] A. Berenstein (B) Department of Mathematics, University of Oregon, Eugene, OR 97403, USA e-mail: [email protected] A. Khoroshkin Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, NY 11794, USA e-mail: [email protected] S. Loktev National Research University Higher School of Economics, 7 Vavilova Str., Moscow, Russia e-mail: [email protected] 586 M. Bennett et al.