Abstract
In this paper, we identify q-deformed \({\mathfrak{gl}_{\ell+1}}\)-Whittaker functions with a specialization of the Macdonald polynomials. This provides a representation of q-deformed \({\mathfrak{gl}_{\ell+1}}\)-Whittaker functions in terms of the Demazure characters of affine Lie algebra \({\widehat{\mathfrak{gl}}_{\ell+1}}\). We also define a system of dual Hamiltonians for q-deformed \({\mathfrak{gl}_{\ell+1}}\)-Toda chains and give a new integral representation for the q-deformed \({\mathfrak{gl}_{\ell+1}}\)-Whittaker functions. Finally, we represent the q-deformed \({\mathfrak{gl}_{\ell+1}}\)-Whittaker function as a matrix element of a quantum torus algebra.
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Awata, H., Odake, S., Shiraishi, J.: Integral representations of the Macdonald symmetric functions. Commun. Math. Phys. 179, 647–666 (1996). arXiv:q-alg/9506006
Casselman W., Shalika J.: The unramified principal series of p-adic groups II. The Whittaker function. Compost. Math. 41, 207–231 (1980)
Cherednik I.: Double Affine Hecke Algebras. Cambridge University Press, Cambridge (2005)
Cheung P., Kac V.: Quantum Calculus. Springer, Heidelberg (2001)
Demazure M.: Désingularisation des variétés de Schubert généralisées. Ann. Sci. Ec. Norm. Supér. 7, 53–88 (1974)
Etingof, P.: Whittaker functions on quantum groups and q-deformed Toda operators. Am. Math. Soc. Transl. Ser.2, 194, 9–25 (1999). arXiv:math.QA/9901053
Fourier, G., Littelmann, P.: Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions. Adv. Math. 211(2), 566–593 (2007). arXiv:math.RT/0509276
Gerasimov, A., Kharchev, S., Lebedev, D., Oblezin, S.: On a Gauss–Givental representation of quantum Toda chain wave function. Int. Math. Res. Notices (2006), Article ID 96489. arXiv:math.QA/0505310
Gerasimov, A., Lebedev, D., Oblezin, S.: On q-deformed \({\mathfrak{gl}_{\ell+1}}\)-Whittaker functions I. Commun. Math.Phys. 294, 97–119 (2010). arXiv:math.RT/0803.0145
Gerasimov, A., Lebedev, D., Oblezin, S.: On q-deformed \({\mathfrak{gl}_{\ell+1} }\)-Whittaker functions II. Commun. Math. Phys. 294, 121–143 (2010). arXiv:math.RT/0803.0970
Ion B.: Nonsymmetric Macdonald polynomials and Demazure characters. Duke Math. J. 116, 299–318 (2003)
Kharchev, S., Lebedev, D.: Eigenfunctions of GL(N,R) Toda chain: the Mellin–Barnes representation. JETP Lett. 71, 235–238 (2000). arXiv:hep-th/0004065
Kharchev, S., Lebedev, D.: M. Semenov–Tian–Shansky, Unitary representations of U q (sl(2,R)), the modular double and the multiparticle q-deformed Toda chains. Commun. Math. Phys. 225, 573–609 (2002). arXiv:hep-th/0102180
Kostant, B.: Quantization and representation theory. In: Proceedings of Symposium on Representation Theory of Lie Groups, London Mathematical Society, vol. 34, pp. 287–317. Lecture Notes Series, Cambridge (1979)
Kumar S.: Demazure character formula in arbitrary Kac–Moody setting. Invent. Math. 89, 395–423 (1987)
Macdonald, I.G.: A new class of symmetric functions, Séminaire Lotharingien de Combinatoire, B20a (1988), 41 pages
Macdonald I.G.: Symmetric Functions and Hall polynomials. Oxford University Press, Oxford (1995)
Mathieu, O.: Formules de charactères pour les algèbres Kac-Moody gènèrales, Astèrisque, pp. 159–160. Society of Mathematics, France, Montrouge (1988)
Ruijsenaars S.N.M.: Relativistic Toda system. Commun. Math. Phys. 133, 217–247 (1990)
Sanderson Y.: On the connection between Macdonald polynomials and Demazure characters. J. Algebraic Combinatorics 11, 269–275 (2000)
Sanderson, Y.: Real characters for Demazure modules of rank two affine Lie algebras. J. Algebra 184 (1996)
Shintani T.: On an explicit formula for class 1 Whittaker functions on GL n over p-adic fields. Proc. Jpn. Acad. 52, 180–182 (1976)
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Gerasimov, A., Lebedev, D. & Oblezin, S. On q-Deformed \({{\mathfrak{gl}}_{\ell+1}}\)-Whittaker Function III. Lett Math Phys 97, 1–24 (2011). https://doi.org/10.1007/s11005-011-0468-y
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DOI: https://doi.org/10.1007/s11005-011-0468-y