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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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