# One-Dimensional Nil-Daha and Whittaker Functions I

- First Online:

- Received:
- Accepted:

DOI: 10.1007/s00031-012-9204-7

- Cite this article as:
- Cherednik, I. & Orr, D. Transformation Groups (2012) 17: 953. doi:10.1007/s00031-012-9204-7

## Abstract

This work is devoted to the theory of nil-DAHA for the root system A_{1} 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 global symmetric function can be interpreted as the generating function of the Demazure characters for dominant weights, which describe the algebraic-geometric properties of the corresponding affine Schubert varieties. Its Harish-Chandra-type asymptotic expansion appears directly related to the solution of the *q*-Toda eigenvalue problem obtained by Givental and Lee in the quantum *K*-theory of ag varieties. It provides an exact mathematical relation between the corresponding physics A-type and B-type models.

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 this work 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 A_{1}.

As the first part of the work, this paper is devoted mainly to the analytic aspects of our construction and the beginning of a systematic algebraic theory of nil-DAHA; the second part will be about the induced modules and their applications to the nonsymmetric global Whittaker functions.