Skip to main content

One-dimensional nil-DAHA and Whittaker functions II

Abstract

This work is devoted to the theory of nil-DAHA for the root system A1 and its applications to symmetric and nonsymmetric (spinor) global q-Whittaker functions, integrating the q-Toda eigenvalue problem and its Dunkl-type nonsymmetric version.

The spinor global functions extend the symmetric ones to the case of all Demazure characters (not only those for dominant weights); the corresponding Gromov–Witten theory is not known. The main result of the paper is a complete algebraic theory of these functions in terms of induced modules of the core subalgebra of nil-DAHA. It is the first instance of the DAHA theory of canonical-crystal bases, quite non-trivial even for A1.

As the second part of this work, this paper is mainly devoted to the theory of the core subalgebra of nil-DAHA, its induced modules and their applications to the nonsymmetric global Whittaker functions. The first part was about the analytic aspects of our construction and a general algebraic theory of nil-DAHA for A1.

This is a preview of subscription content, access via your institution.

References

  1. [Ch1]

    I. Cherednik, Difference Macdonald–Mehta conjecture, IMRN 10 (1997), 449–467.

    MathSciNet  Article  Google Scholar 

  2. [Ch2]

    I. Cherednik, Toward Harmonic Analysis on DAHA (Integral formulas for canonical traces), notes of the lecture delivered at University of Amsterdam (May 30, 2008), http://math.mit.edu/~etingof/hadaha.pdf.

  3. [ChM]

    I. Cherednik, X. Ma, Spherical and Whittaker functions via DAHA I, II, arXiv:0904.4324 (2009), to be published by Selecta Mathematica.

  4. [ChO]

    I. Cherednik, D. Orr, One-dimensional nil-DAHA and Whittaker functions I, Transform. Groups 17 (2012), no. 4, 953–987.

    MathSciNet  MATH  Article  Google Scholar 

  5. [O]

    E. Opdam, On the spectral decomposition of affine Hecke algebras, J. Inst. Math. Jussieu 3 (2004), no. 4, 531–648.

    MathSciNet  MATH  Article  Google Scholar 

  6. [Sto]

    J. Stokman, Difference Fourier transforms for nonreduced root systems, Selecta Math. (N.S.) 9 (2003), no. 3, 409–494.

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Ivan Cherednik.

Additional information

*Partially supported by NSF grant DMS–1101535.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Cherednik, I., Orr, D. One-dimensional nil-DAHA and Whittaker functions II. Transformation Groups 18, 23–59 (2013). https://doi.org/10.1007/s00031-013-9210-4

Download citation

Keywords

  • Dirac Operator
  • Spinor Operator
  • Polynomial Representation
  • WHITTAKER Function
  • Dunkl Operator