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Nonsymmetric Macdonald polynomials and Demazure characters. (English) Zbl 1039.33008
The author studies the specialization $$E_\lambda(q,\infty)$$ of the nonsymmetric Macdonald polynomials $$E_\lambda(q,t)$$ associated with an irreducible affine root system for which the affine simple root is short. He establishes a connection between $$E_\lambda(q,\infty)$$ and the Demazure characters of the corresponding affine Kac-Moody algebra. As a consequence, he derives a representation-theoretical interpretation of the coefficients of the expansion of the symmetric Macdonald polynomials in the basis formed by the irreducible characters of the associated finite Lie algebra. Moreover, he relates the specialization $$E_\lambda(\infty,\infty)$$ of $$E_\lambda(q,t)$$ to the Demazure characters of the finite irreducible Lie algebra.
This paper extends the results obtained by Y. B. Sanderson [J. Algebr. Comb. 11, No. 3, 269–275 (2000; Zbl 0957.05106)] in the case of an irreducible root system of type $$A_n$$. The proofs rely on the method of intertwiners in double affine Hecke algebras.

##### MSC:
 33D52 Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.) 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 20C08 Hecke algebras and their representations 33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
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##### References:
 [1] I. Cherednik, Double affine Hecke algebras and Macdonald’s conjectures , Ann. of Math. (2) 141 (1995), 191–216. JSTOR: · Zbl 0822.33008 [2] –. –. –. –., Nonsymmetric Macdonald polynomials , Internat. Math. Res. Notices 1995 , 483–515. · Zbl 0886.05121 [3] –. –. –. –., Intertwining operators of double affine Hecke algebras , Selecta Math. (N.S.) 3 (1997), 459–495. · Zbl 0907.20042 [4] J. E. Humphreys, Reflection Groups and Coxeter Groups , Cambridge Stud. Adv. Math. 29 , Cambridge Univ. Press, Cambridge, 1990. · Zbl 0725.20028 [5] B. Ion, Involutions of double affine Hecke algebras , to appear in Compositio Math., · Zbl 1062.20005 [6] V. G. Kac, Infinite-dimensional Lie algebras , 3d ed., Cambridge Univ. Press, Cambridge, 1990. · Zbl 0716.17022 [7] F. Knop, Integrality of two variable Kostka functions , J. Reine Angew. Math. 482 (1997), 177–189. · Zbl 0876.05098 [8] F. Knop and S. Sahi, A recursion and a combinatorial formula for Jack polynomials , Invent. Math. 128 (1997), 9–22. · Zbl 0870.05076 [9] S. Kumar, Demazure character formula in arbitrary Kac-Moody setting , Invent. Math. 89 (1987), 395–423. · Zbl 0635.14023 [10] G. Lusztig, Green polynomials and singularities of unipotent classes , Adv. Math. 42 (1981), 169–178. · Zbl 0473.20029 [11] –. –. –. –., Affine Hecke algebras and their graded version , J. Amer. Math. Soc. 2 (1989), 599–635. JSTOR: · Zbl 0715.22020 [12] I. G. Macdonald, Affine Hecke algebras and orthogonal polynomials , Astérisque 237 (1996), 189–207., Séminaire Bourbaki 1994/95, exp. no. 797. · Zbl 0883.33008 [13] O. Mathieu, Formules de caractères pour les algèbres de Kac-Moody générales , Astérisque 159 – 160 , Soc. Math. France, Montrouge, 1988. · Zbl 0683.17010 [14] E. M. Opdam, Harmonic analysis for certain representations of graded Hecke algebras , Acta Math. 175 (1995), 75–121. · Zbl 0836.43017 [15] S. Sahi, Interpolation, integrality, and a generalization of Macdonald’s polynomials , Internat. Math. Res. Notices 1996 , 457–471. · Zbl 0861.05063 [16] –. –. –. –., Nonsymmetric Koornwinder polynomials and duality , Ann. of Math. (2) 150 (1999), 267–282. JSTOR: · Zbl 0941.33013 [17] –. –. –. –., A new formula for weight multiplicities and characters , Duke Math. J. 101 (2000), 77–84. · Zbl 0953.17003 [18] –. –. –. –., “Some properties of Koornwinder polynomials” in $$q$$-Series from a Contemporary Perspective (South Hadley, Mass., 1998) , Contemp. Math. 254 , Amer. Math. Soc., Providence, 2000, 395–411. · Zbl 0959.33008 [19] Y. B. Sanderson, On the connection between Macdonald polynomials and Demazure characters , J. Algebraic Combin. 11 (2000), 269–275. · Zbl 0957.05106
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