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.
 A. Braverman, M. Finkelberg, Semiinfinite Schubert varieties and quantum Ktheory of flag manifolds, preprint arXiv:1111.2266 (2011).
 I. Cherednik, Double Affine Hecke algebras, London Mathematical Society Lecture Note Series, Vol. 319, Cambridge University Press, Cambridge, 2006.
 I. Cherednik, Intertwining operators of double affine Hecke algebras, Selecta Math., New ser. 3 (1997), 459–495. CrossRef
 I. Cherednik, Difference Macdonald−Mehta conjecture, IMRN 10 (1997), 449–467. CrossRef
 I. Cherednik, Toward harmonic analysis on DAHA (Integral formulas for canonical traces), notes of the lecture delivered at the University of Amsterdam (May 30, 2008), http://math.mit.edu~/etingof/hadaha.pdf.
 I. Cherednik, Whittaker limits of difference spherical functions, IMRN 20 (2009), 3793–3842, arXiv:0807.2155 (2008).
 I. Cherednik, Affine extensions of Knizhnik−Zamolodchikov equations and Lusztig’s isomorphisms, in: Special Functions, Proceedings of the Hayashibara forum 1990, Okayama, SpringerVerlag, 1991, pp. 63–77.
 I. Cherednik, X. Ma, Spherical and Whittaker functions via DAHA I, II, preprint arXiv:0904.4324 (2009), to be published by Selecta Mathematica.
 I. Cherednik, D. Orr, Onedimensional nilDAHA and Whittaker functions II, to appear in Transformation Groups.
 P. Etingof, Whittaker functions on quantum groups and qdeformed Toda operators, in: Differential Topology, InfiniteDimensional Lie Algebras, and Applications, AMS Transl. Ser. 2, Vol. 194, AMS, Providence, Rhode Island, 1999, pp. 9–25.
 B. Feigin, E. Feigin, M. Jimbo, T. Miwa, E. Mukhin, Fermionic formulas for eigenfunctions of the difference Toda Hamiltonian, Lett. Math. Phys. 88 (2009), 39–77. CrossRef
 A. Gerasimov, D. Lebedev, S. Oblezin, On qdeformed \( \mathfrak{g}{{\mathfrak{l}}_{l+1 }} \) Whittaker function III, Lett. Math. Phys. 97 (2011), no. 1, 1–24. CrossRef
 A. Givental, Y. P. Lee, Quantum Ktheory on flag manifolds, finitedifference Toda lattices and quantum groups, Inventiones Math. 151 (2003), 193–219. CrossRef
 R. Goodman, N. R. Wallach, Conical vectors and Whittaker vectors, J. Functional Analysis 39 (1980), 199–279. CrossRef
 G. J. Heckman, E. M. Opdam, Root systems and hypergeometric functions I, Comp. Math. 64 (1987), 329–352.
 B. Ion, Nonsymmetric Macdonald polynomials and Demazure characters, Duke Math. J. 116 (2003), no. 2, 299–318. CrossRef
 B. Kostant, S. Kumar, TEquivariant Ktheory of generalized flag varieties, J. Diff. Geom. 32 (1990), 549–603.
 D. Lubinsky, On qexponential functions for q = 1, Canad. Math. Bull. 41 (1998), no. 1, 86–97. CrossRef
 I. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Oxford University Press, 1999.
 M. van Meer, Bispectral quantum Knizhnik−Zamolodchikov equations for arbitrary root systems, Selecta Math. (N.S.) 17 (2011), no. 1, 183–221. CrossRef
 M. van Meer, J. V. Stokman, Double affine Hecke algebras and bispectral quantum Knizhnik−Zamolodchikov equations, Int. Math. Res. Not. 6, (2010) 969–1040.
 S. Ruijsenaars, Systems of Calogero−Moser type, in: Proceedings of the 1994 Banff summer school “Particles and Fields”, G. Semenoff, L. Vinet, eds., CRM Ser. in Math. Phys., Springer, New York, 1999, pp. 251–352.
 Y. Sanderson, On the Connection between Macdonald polynomials and Demazure characters, J. Alg. Comb. 11 (2000), 269–275. CrossRef
 J. Stokman, The cfunction expansion of a basic hypergeometric function associated to root systems, preprint arXiv:1109.0613 (2011).
 S. Suslov, Another addition theorem for the qexponential function, J. Phys. A: Math. Gen. 33 (2000), no. 41, L375−L380. CrossRef
 K. Taipale, Ktheoretic Jfunctions of type A flag varieties, preprint arXiv:1110.3117 (2011).
 N. R. Wallach, Real Reductive Groups II, Academic Press, Boston, 1992.
 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