OneDimensional NilDaha and Whittaker Functions I
 Ivan Cherednik,
 Daniel Orr
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This work is devoted to the theory of nilDAHA for the root system A _{1} and its applications to symmetric and nonsymmetric (spinor) global qWhittaker functions, integrating the qToda eigenvalue problem and its Dunkltype nonsymmetric version.
The global symmetric function can be interpreted as the generating function of the Demazure characters for dominant weights, which describe the algebraicgeometric properties of the corresponding affine Schubert varieties. Its HarishChandratype asymptotic expansion appears directly related to the solution of the qToda eigenvalue problem obtained by Givental and Lee in the quantum Ktheory of ag varieties. It provides an exact mathematical relation between the corresponding physics Atype and Btype 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 nilDAHA. It is the first instance of the DAHA theory of canonicalcrystal bases, quite nontrivial 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 nilDAHA; the second part will be about the induced modules and their applications to the nonsymmetric global Whittaker functions.
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 Title
 OneDimensional NilDaha and Whittaker Functions I
 Journal

Transformation Groups
Volume 17, Issue 4 , pp 953987
 Cover Date
 20121201
 DOI
 10.1007/s0003101292047
 Print ISSN
 10834362
 Online ISSN
 1531586X
 Publisher
 SP Birkhäuser Verlag Boston
 Additional Links
 Authors

 Ivan Cherednik ^{(1)}
 Daniel Orr ^{(1)}
 Author Affiliations

 1. Department of Mathematics, University of North Carolina, Chapel Hill, NC, 27599, USA