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Nonsymmetric difference Whittaker functions

Abstract

Starting with nonsymmetric global difference spherical functions, we define and calculate spinor (nonsymmetric) global \(q\)-Whittaker functions for arbitrary reduced root systems, which are reproducing kernels of the DAHA-Fourier transforms of Nil-DAHA and solutions of the \(q\)-Toda–Dunkl eigenvalue problem. We introduce the spinor \(q\)-Toda–Dunkl operators as limits of the difference Dunkl operators in DAHA theory under the spinor variant of the Ruijsenaars procedure. Their general algebraic theory (any reduced root systems) is the key part of this paper, based on the new technique of \(W\)-spinors and corresponding developments in combinatorics of affine root systems.

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Acknowledgments

The first author thanks Tetsuji Miwa and the Mathematics Department of Kyoto University (where the paper was started) for the invitation and hospitality, as well as IHES (where the paper was completed). We thank Eric Opdam for useful remarks and Evgeny Feigin for his help with Conjecture 3.7. The second author thanks the organizers of the 5th Southeast Lie Theory Workshop for the invitation, where the results of this paper were reported.

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Correspondence to Ivan Cherednik.

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Ivan Cherednik is partially supported by NSF Grant DMS-1101535.

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Cherednik, I., Orr, D. Nonsymmetric difference Whittaker functions. Math. Z. 279, 879–938 (2015). https://doi.org/10.1007/s00209-014-1397-0

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Keywords

  • Root systems
  • Hecke algebras
  • Whittaker functions
  • Toda operators
  • Macdonald polynomials

Mathematics Subject Classification

  • 20C08
  • 22E35
  • 33D80
  • 22E66
  • 20G44