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Nonsymmetric Macdonald polynomials and PBW filtration: towards the proof of the Cherednik-Orr conjecture. (English) Zbl 1319.05141
Summary: The Cherednik-Orr conjecture expresses the \(t \to \infty\) limit of the nonsymmetric Macdonald polynomials in terms of the PBW twisted characters of the affine level one Demazure modules. We prove this conjecture in several special cases.

MSC:
05E10 Combinatorial aspects of representation theory
05E05 Symmetric functions and generalizations
33D52 Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.)
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[1] Bourbaki, N., Éléments de mathématique, fasc. XXXIV, groupes et algèbres de Lie, Actualités Scientifiques et Industrielles, vol. 1337, (1968), Hermann Paris, Chapitres IV, V, VI · Zbl 0186.33001
[2] Chari, V.; Loktev, S., Weyl, Demazure and fusion modules for the current algebra of \(\mathfrak{sl}_{r + 1}\), Adv. Math., 207, 928-960, (2006) · Zbl 1161.17318
[3] Cherednik, I., Nonsymmetric Macdonald polynomials, Int. Math. Res. Not. IMRN, 10, 483-515, (1995) · Zbl 0886.05121
[4] Cherednik, I., Double affine Hecke algebras, London Mathematical Society Lecture Note Series, vol. 319, (2006), Cambridge University Press Cambridge · Zbl 1097.20007
[5] Cherednik, I.; Feigin, E., Extremal part of the PBW-filtration and E-polynomials · Zbl 1378.17040
[6] Cherednik, I.; Orr, D., Nonsymmetric difference Whittaker functions, (2013), preprint · Zbl 1372.20009
[7] Cherednik, I.; Orr, D., One-dimensional nil-DAHA and Whittaker functions, Transform. Groups, 18, 1, 23-59, (2013) · Zbl 1320.20004
[8] Feigin, E., The PBW filtration, Demazure modules and toroidal current algebras, SIGMA Symmetry Integrability Geom. Methods Appl., 4, 070, (2008), 21 pp · Zbl 1215.17015
[9] Feigin, E., \(\mathbb{G}_a^M\) degeneration of flag varieties, Selecta Math. (N.S.), 18, 3, 513-537, (2012) · Zbl 1267.14064
[10] Feigin, E.; Fourier, G.; Littelmann, P., PBW-filtration and bases for irreducible modules in type \(A_n\), Transform. Groups, 16, 1, 71-89, (2011) · Zbl 1237.17011
[11] Feigin, E.; Fourier, G.; Littelmann, P., PBW filtration and bases for symplectic Lie algebras, Int. Math. Res. Not. IMRN, 24, 5760-5784, (2011) · Zbl 1233.17007
[12] Feigin, E.; Fourier, G.; Littelmann, P., PBW-filtration over \(\mathbb{Z}\) and compatible bases for \(V_{\mathbb{Z}}(\lambda)\) in type \(A_n\) and \(C_n\), Symmetries, Integrable Systems and Representations, vol. 40, 35-63, (2013), Springer · Zbl 1323.17011
[13] Fourier, G.; Littelmann, P., Tensor product structure of affine Demazure modules and limit constructions, Nagoya Math. J., 182, 171-198, (2006) · Zbl 1143.22010
[14] Fourier, G.; Littelmann, P., Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions, Adv. Math., 211, 2, 566-593, (2007) · Zbl 1114.22010
[15] Gornitsky, A., Essential signatures and canonical bases in irreducible representations of the group \(G_2\), (2011), (in Russian)
[16] Haiman, M.; Haglund, J.; Loehr, N., A combinatorial formula for non-symmetric Macdonald polynomials, Amer. J. Math., 130, 2, 359-383, (2008) · Zbl 1246.05162
[17] Ion, B., Nonsymmetric Macdonald polynomials and Demazure characters, Duke Math. J., 116, 2, 299-318, (2003) · Zbl 1039.33008
[18] Kac, V., Infinite dimensional Lie algebras, (1990), Cambridge University Press Cambridge · Zbl 0716.17022
[19] Knop, F., Integrality of two variable kostka functions, J. Reine Angew. Math., 482, 177-189, (1997) · Zbl 0876.05098
[20] Knop, F.; Sahi, S., A recursion and a combinatorial formula for Jack polynomials, Invent. Math., 128, 1, 9-22, (1997) · Zbl 0870.05076
[21] Kumar, S., Kac-Moody groups, their flag varieties and representation theory, Progress in Mathematics, vol. 204, (2002), Birkhäuser Boston, Inc. Boston, MA · Zbl 1026.17030
[22] Kus, D.; Littelmann, P., Fusion products and toroidal algebras · Zbl 1381.17014
[23] Macdonald, I. G., Symmetric functions and Hall polynomials, (1995), Oxford University Press · Zbl 0487.20007
[24] I.G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Séminaire Bourbaki, vol. 1994/95, Astérisque No. 237 (1996), Exp. No. 797, 4, 189-207. · Zbl 0883.33008
[25] Opdam, E., Harmonic analysis for certain representations of graded Hecke algebras, Acta Math., 175, 1, 75-121, (1995) · Zbl 0836.43017
[26] Orr, D.; Shimozono, M., Specializations of nonsymmetric Macdonald-Koornwinder polynomials · Zbl 1381.05089
[27] Sanderson, Y., On the connection between Macdonald polynomials and Demazure characters, J. Algebraic Combin., 11, 269-275, (2000) · Zbl 0957.05106
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