Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-13T05:13:27.679Z Has data issue: false hasContentIssue false
  • English
  • Français

BGG reciprocity for current algebras

Part of: Lie algebras and Lie superalgebras Basic hypergeometric functions

Published online by Cambridge University Press:  09 February 2015

Vyjayanthi Chari  and
Bogdan Ion
Vyjayanthi Chari
Affiliation:
Department of Mathematics, University of California, Riverside, CA 92521, USA email chari@math.ucr.edu
Bogdan Ion
Affiliation:
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA email bion@pitt.edu Algebra and Number Theory Research Center, Faculty of Mathematics and Computer Science, University of Bucharest, 14 Academiei St., Bucharest, Romania
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In Bennett et al. [BGG reciprocity for current algebras, Adv. Math. 231 (2012), 276–305] it was conjectured that a BGG-type reciprocity holds for the category of graded representations with finite-dimensional graded components for the current algebra associated to a simple Lie algebra. We associate a current algebra to any indecomposable affine Lie algebra and show that, in this generality, the BGG reciprocity is true for the corresponding category of representations.

Keywords

current algebraWeyl moduleMacdonald polynomial

MSC classification

Primary: 17B10: Representations, algebraic theory (weights) 17B67: Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 33D52: Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.)
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 7 , July 2015 , pp. 1265 - 1287
Copyright
© The Authors 2015 

References

Bennet, M., Berenstein, A., Chari, V., Khoroshkin, A. and Loktev, S., Macdonald polynomials and BGG reciprocity for current algebras, Selecta Math. 20 (2014), 585–607.CrossRefGoogle Scholar
Bennett, M., Chari, V. and Manning, N., BGG reciprocity for current algebras, Adv. Math. 231 (2012), 276–305.CrossRefGoogle Scholar
Bernstein, I. N., Gelfand, I. M. and Gelfand, S. I., A certain category of g-modules, Funktsional. Anal. i Prilozhen. 10 (1976), 1–8.Google Scholar
Chari, V., Integrable representations of affine Lie-algebras, Invent. Math. 85 (1986), 317–335.CrossRefGoogle Scholar
Chari, V., Fourier, G. and Khandai, T., A categorical approach to Weyl modules, Transform. Groups 15 (2010), 517–549.CrossRefGoogle Scholar
Chari, V., Fourier, G. and Senesi, P., Weyl modules for the twisted loop algebras, J. Algebra 319 (2008), 5016–5038.CrossRefGoogle Scholar
Chari, V. and Greenstein, J., Current algebras, highest weight categories and quivers, Adv. Math. 216 (2007), 811–840.CrossRefGoogle Scholar
Chari, V., Ion, B. and Kus, D., Weyl modules for the hyperspecial current algebra, Int. Math. Res. Not. IMRN (2014), doi:10.1093/imrn/rnu135; arXiv:1403.5285.Google Scholar
Chari, V. and Loktev, S., Weyl, Demazure and fusion modules for the current algebra of slr+1, Adv. Math. 207 (2006), 928–960.CrossRefGoogle Scholar
Chari, V. and Pressley, A., Weyl modules for classical and quantum affine algebras, Represent. Theory 5 (2001), 191–223.CrossRefGoogle Scholar
Cherednik, I., Double affine Hecke algebras and Macdonald’s conjectures, Ann. of Math. (2) 141 (1995), 191–216.CrossRefGoogle Scholar
Cherednik, I., Macdonald’s evaluation conjectures and difference Fourier transform, Invent. Math. 122 (1995), 119–145.CrossRefGoogle Scholar
Fourier, G., Manning, N. and Savage, A., Global Weyl modules for equivariant map algebras, Int. Math. Res. Not. IMRN (2014), doi:10.1093/imrn/rnt231.CrossRefGoogle Scholar
Fourier, G. and Kus, D., Demazure modules and Weyl modules: the twisted current case, Trans. Amer. Math. Soc. 365 (2013), 6037–6064.CrossRefGoogle Scholar
Fourier, G. and Littelmann, P., Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions, Adv. Math. 211 (2007), 566–593.CrossRefGoogle Scholar
Gustafson, R. A., A generalization of Selberg’s beta integral, Bull. Amer. Math. Soc. (N.S.) 22 (1990), 97–105.CrossRefGoogle Scholar
Ion, B., Nonsymmetric Macdonald polynomials and Demazure characters, Duke Math. J. 116 (2003), 299–318.CrossRefGoogle Scholar
Kac, V., Infinite-dimensional Lie algebras, third edition (Cambridge University Press, Cambridge, 1990).Google Scholar
Koornwinder, T. H., Askey–Wilson polynomials for root systems of type BC, in Hypergeometric functions on domains of positivity, Jack polynomials, and applications (Tampa, FL, 1991), Contemporary Mathematics, vol. 138 (American Mathematical Society, Providence, RI, 1992), 189–204.CrossRefGoogle Scholar
Lenart, C., From Macdonald polynomials to a charge statistic beyond type A, J. Combin. Theory Ser. A 119 (2012), 683–712.CrossRefGoogle Scholar
Lenart, C. and Lubovsky, A., A generalization of the alcove model and its applications, in 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012), Discrete Mathematics & Theoretical Computer Science Proceedings (DMTCS, Nancy, 2012), 875–886.Google Scholar
Lenart, C., Naito, S., Sagaki, D., Schilling, A. and Shimozono, M., A uniform model for Kirillov–Reshetikhin crystals II. Alcove model, path model, and $P=X$, Preprint (2014),arXiv:1402.220.Google Scholar
Macdonald, I. G., Orthogonal polynomials associated with root systems, Sém. Lothar. Combin. 45 (2000/01), Art. B45a.Google Scholar
Macdonald, I. G., Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, vol. 157 (Cambridge University Press, Cambridge, 2003).CrossRefGoogle Scholar
Naoi, K., Weyl modules, Demazure modules and finite crystals for non-simply laced type, Adv. Math. 229 (2012), 875–934.CrossRefGoogle Scholar
Neher, E., Savage, A. and Senesi, P., Irreducible finite-dimensional representations of equivariant map algebras, Trans. Amer. Math. Soc. 364 (2012), 2619–2646.CrossRefGoogle Scholar
Ram, A. and Yip, M., A combinatorial formula for Macdonald polynomials, Adv. Math. 226 (2011), 309–331.CrossRefGoogle Scholar
Quillen, D., Projective modules over polynomial rings, Invent. Math. 36 (1976), 167–171.CrossRefGoogle Scholar
Sahi, S., Nonsymmetric Koornwinder polynomials and duality, Ann. of Math. (2) 150 (1999), 267–282.CrossRefGoogle Scholar
Sanderson, Y. B., On the connection between Macdonald polynomials and Demazure characters, J. Algebraic Combin. 11 (2000), 269–275.CrossRefGoogle Scholar
Stokman, J. V., Koornwinder polynomials and affine Hecke algebras, Int. Math. Res. Not. IMRN 2000 (2000), 1005–1042.CrossRefGoogle Scholar
Suslin, A. A., Projective modules over polynomial rings are free, Dokl. Akad. Nauk SSSR 229 (1976), 1063–1066 (in Russian).Google Scholar
Tits, J., Reductive groups over local fields, in Automorphic forms, representations and L-functions (Corvallis, OR, 1977), Part 1, Proceedings of Symposia in Pure Mathematics, vol. XXXIII (American Mathematical Society, Providence, RI, 1979), 29–69.CrossRefGoogle Scholar
van Diejen, J. F., Self-dual Koornwinder–Macdonald polynomials, Invent. Math. 126 (1996), 319–339.CrossRefGoogle Scholar