One-Dimensional Nil-Daha and Whittaker Functions I Authors Ivan Cherednik Department of Mathematics University of North Carolina Daniel Orr Department of Mathematics University of North Carolina Article

First Online: 16 November 2012 Received: 05 April 2012 Accepted: 09 September 2012 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.

Partially supported by NSF grant DMS−1101535.

References [BL]

A. Braverman, M. Finkelberg,

Semi-infinite Schubert varieties and quantum K-theory of flag manifolds , preprint

arXiv:1111.2266 (2011).

[Ch1]

I. Cherednik, Double Affine Hecke algebras , London Mathematical Society Lecture Note Series, Vol. 319, Cambridge University Press, Cambridge, 2006.

[Ch2]

I. Cherednik,

Intertwining operators of double affine Hecke algebras , Selecta Math., New ser.

3 (1997), 459–495.

MathSciNet MATH CrossRef [Ch3]

I. Cherednik,

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

MathSciNet CrossRef [Ch4]

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 .

[Ch5]

I. Cherednik,

Whittaker limits of difference spherical functions , IMRN

20 (2009), 3793–3842,

arXiv:0807.2155 (2008).

[Ch6]

I. Cherednik, Affine extensions of Knizhnik−Zamolodchikov equations and Lusz-tig’s isomorphisms , in: Special Functions , Proceedings of the Hayashibara forum 1990, Okayama, Springer-Verlag, 1991, pp. 63–77.

[ChM]

I. Cherednik, X. Ma,

Spherical and Whittaker functions via DAHA I, II, preprint

arXiv:0904.4324 (2009), to be published by Selecta Mathematica.

[ChO]

I. Cherednik, D. Orr, One-dimensional nil-DAHA and Whittaker functions II, to appear in Transformation Groups.

[Et]

P. Etingof, Whittaker functions on quantum groups and q-deformed Toda operators , in: Differential Topology, Infinite-Dimensional Lie Algebras, and Applications , AMS Transl. Ser. 2, Vol. 194, AMS, Providence, Rhode Island, 1999, pp. 9–25.

[FJM]

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.

MathSciNet MATH CrossRef [GLO]

A. Gerasimov, D. Lebedev, S. Oblezin,

On q-deformed
\( \mathfrak{g}{{\mathfrak{l}}_{l+1 }} \) -

Whittaker function III, Lett. Math. Phys.

97 (2011), no. 1, 1–24.

MathSciNet MATH CrossRef [GiL]

A. Givental, Y. P. Lee,

Quantum K-theory on flag manifolds, finite-difference Toda lattices and quantum groups , Inventiones Math.

151 (2003), 193–219.

MathSciNet MATH CrossRef [GW]

R. Goodman, N. R. Wallach,

Conical vectors and Whittaker vectors , J. Functional Analysis

39 (1980), 199–279.

MathSciNet MATH CrossRef [HO]

G. J. Heckman, E. M. Opdam,

Root systems and hypergeometric functions I, Comp. Math.

64 (1987), 329–352.

MathSciNet MATH [Ion]

B. Ion,

Nonsymmetric Macdonald polynomials and Demazure characters , Duke Math. J.

116 (2003), no. 2, 299–318.

MathSciNet MATH CrossRef [KK]

B. Kostant, S. Kumar,

T-Equivariant K-theory of generalized flag varieties , J. Diff. Geom.

32 (1990), 549–603.

MathSciNet MATH [LuB]

D. Lubinsky,

On q-exponential functions for |

q | = 1, Canad. Math. Bull.

41 (1998), no. 1, 86–97.

MathSciNet MATH CrossRef [Ma]

I. Macdonald, Symmetric Functions and Hall Polynomials , 2nd ed., Oxford University Press, 1999.

[Me]

M. van Meer,

Bispectral quantum Knizhnik−Zamolodchikov equations for arbitrary root systems , Selecta Math. (N.S.)

17 (2011), no. 1, 183–221.

MathSciNet MATH CrossRef [MS]

M. van Meer, J. V. Stokman, Double affine Hecke algebras and bispectral quantum Knizhnik−Zamolodchikov equations , Int. Math. Res. Not. 6 , (2010) 969–1040.

[Rui]

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.

[San]

Y. Sanderson,

On the Connection between Macdonald polynomials and Demazure characters , J. Alg. Comb.

11 (2000), 269–275.

MathSciNet MATH CrossRef [Sto]

J. Stokman,

The c-function expansion of a basic hypergeometric function associated to root systems , preprint

arXiv:1109.0613 (2011).

[Sus]

S. Suslov,

Another addition theorem for the q-exponential function , J. Phys. A: Math. Gen.

33 (2000), no. 41, L375−L380.

MathSciNet MATH CrossRef [T]

K. Taipale,

K-theoretic J-functions of type A flag varieties , preprint

arXiv:1110.3117 (2011).

[Wa]

N. R. Wallach,

Real Reductive Groups II, Academic Press, Boston, 1992.

MATH © Springer Science+Business Media New York 2012