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Specializations of nonsymmetric Macdonald–Koornwinder polynomials

Abstract

The purpose of this article is to work out the details of the Ram–Yip formula for nonsymmetric Macdonald–Koornwinder polynomials for the double affine Hecke algebras of not-necessarily reduced affine root systems. It is shown that the \(t\rightarrow 0\) equal-parameter specialization of nonsymmetric Macdonald polynomials admits an explicit combinatorial formula in terms of quantum alcove paths, generalizing the formula of Lenart in the untwisted case. In particular, our formula yields a definition of quantum Bruhat graph for all affine root systems. For mixed type, the proof requires the Ram–Yip formula for the nonsymmetric Koornwinder polynomials. A quantum alcove path formula is also given at \(t\rightarrow \infty \). As a consequence, we establish the positivity of the coefficients of nonsymmetric Macdonald polynomials under this limit, as conjectured by Cherednik and the first author. Finally, an explicit formula is given at \(q\rightarrow \infty \), which yields the p-adic Iwahori–Whittaker functions of Brubaker, Bump, and Licata.

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Notes

  1. 1.

    In [3], the authors use \(E_\lambda (X;q^{-1};t^{-1})\) and correspondingly send \(q\rightarrow 0\); [19] also uses this convention for q and t.

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Acknowledgements

Thanks to Bogdan Ion, Arun Ram, and Siddartha Sahi, for patient explanations about the double affine Hecke algebra. Thanks to Anne Schilling and Nicolas Thiéry for implementing nonsymmetric Macdonald polynomials in sage [31]; we used their program extensively. Thanks to ICERM, which provided the venue for the above activities. Thanks to Cristian Lenart, Satoshi Naito, Daisuke Sagaki, and Anne Schilling for related collaborations. Thanks to Ivan Cherednik for helpful discussions. The second author thanks the NSF for the support from Grant NSF DMS-1200804, and both authors thank the NSF for partial support from Grant NSF DMS-1600653.

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Correspondence to Daniel Orr.

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Orr, D., Shimozono, M. Specializations of nonsymmetric Macdonald–Koornwinder polynomials. J Algebr Comb 47, 91–127 (2018). https://doi.org/10.1007/s10801-017-0770-6

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Keywords

  • Macdonald-Koornwinder polynomials
  • Double affine Hecke algebras
  • Alcove paths
  • Quantum Bruhat graph