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One-dimensional nil-DAHA and Whittaker functions II


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.

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Correspondence to Ivan Cherednik.

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*Partially supported by NSF grant DMS–1101535.

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Cherednik, I., Orr, D. One-dimensional nil-DAHA and Whittaker functions II. Transformation Groups 18, 23–59 (2013).

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