On q-Deformed gll+1-Whittaker Function I

Communications in Mathematical Physics

Volumn 294, Issue 1, 2009, Pages 97-119

  Gerasimov, Anton ^a,b   Lebedev, Dimitri ^a   Oblezin, Sergey ^a  

^a INSTITUTE FOR THEORETICAL AND EXPERIMENTAL PHYSICS   (Russian Federation)

^b TRINITY COLLEGE DUBLIN   (Ireland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 72249087789     PISSN: 00103616     EISSN: 14320916     Source Type: Journal    
DOI: 10.1007/s00220-009-0917-y     Document Type: Article

References (29)

1. Awata H., Odake S., Shiraishi J.: Integral representations of the Macdonald symmetric functions. Commun. Math. Phys. 179, 647-666 (1996).
2. Braverman A., Finkelberg M.: Finite-difference quantum Toda lattice via equivariant K-theory. Trans. Groups 10, 363-386 (2005).
3. Casselman W., Shalika J.: The unramified principal series of p-adic groups II. The Whittaker function. Comp. Math. 41, 207-231 (1980).
4. Cherednik, I. V.: Quantum groups as hidden symmetries of classic representation theory. In Differential Geometric Methods in Theoretical Physics (Chester, 1988), Teaneck, NJ: World Sci. Publishing, 1989, pp. 47-54.
5. Cheung P., Kac V.: Quantum Calculus. Springer, Berlin-Heidelberg-New York (2001).
6. Etingof, P.: Whittaker functions on quantum groups and q-deformed Toda operators. Amer. Math. Soc. Transl. Ser. 2, Vol. 194, Providence, RI: Amer. Math. Soc., 1999, pp. 9-25.
7. Etingof P. I., Kirillov A. A. Jr.: Macdonald's polynomials and Representations of Quantum Groups. Math. Res. Let. 1, 279-296 (1994).
8. Inozemtsev V. I.: Finite Toda lattice. Commun. Math. Phys. 121, 629-638 (1989).
9. Fulton W., Harris J.: Representation Theory. Springer, A First Course. Berlin-Heidelberg-New York (1991).
10. 3 subspaces and quantum Toda hamiltonians. http://arxiv. org/abs/0707. 1635v2[math. QA], 2007.
11. Gerasimov, A., Kharchev, S., Lebedev, D.: Representation Theory and Quantum Inverse Scattering Method: Open Toda Chain and Hyperbolic Sutherland Model, Int. Math. Res. Notes, No. 17, 823-854 (2004).
12. Gerasimov, A., Kharchev, S., Lebedev, D.: Representation Theory and Quantum Integrability. Progress in Math. 237, Basel: Birkhäuser, 2005, pp. 133-156.
13. Gerasimov, A., Kharchev, S., Lebedev, D., Oblezin, S.: On a Gauss-Givental representation of quantum Toda chain wave function. Int. Math. Res. Notices, ArticleID 96489, 23 pages, 2006.
14. Gerasimov A., Lebedev D., Oblezin S.: Baxter operator and Archimedean Hecke algebra. Commun. Math. Phys. 284(3), 867-896 (2008).
15. Gerasimov, A., Lebedev, D., Oblezin, S.: New Integral Representations of Whittaker Functions for Classical Lie Groups. http://arxiv. org/abs/0705. 2886v1[math. RT], 2007.
16. l+1-Whittaker functions II. Commun. Math. Phys., to apper (this issue), 2009, http://arxiv. org/abs/0805. 3754v2[math. RT], 2008.
17. Gerasimov A., Lebedev D., Oblezin S.: Baxter Q-operators and their Arithmetic implications. Lett. Math. Phys. 88(1-3), 3-30 (2009).
18. Givental A., Kim B.: Quantum cohomology of flag manifolds and Toda lattices. Commun. Math. Phys. 168, 609-641 (1995).
19. Givental, A.: Stationary Phase Integrals, Quantum Toda Lattices, Flag Manifolds and the Mirror Conjecture. In: Topics in Singularity Theory, Amer. Math. Soc. Transl. Ser., 2 Vol. 180, Providence, RI: Amer. Math. Soc., 1997, pp. 103-115.
20. Givental A., Lee Y.-P.: Quantum K-theory on flag manifolds, finite-difference Toda lattices and quantum groups. Invent. Math. 151, 193-219 (2003).
21. Kirillov, A. Jr.: Traces of intertwining operators and Macdonald's polynomials. PhD Thesis, Yale University, May 1995, available at http://arxiv. org/abs/q-alg/9503012v1, 1995.
22. Kharchev S., Lebedev D.: Eigenfunctions of GL(N, R) Toda chain: The Mellin-Barnes representation. JETP Lett. 71, 235-238 (2000).
23. Kharchev S., Lebedev D., Semenov-Tian-Shansky M.: Unitary representations of Uq(sl(2, R)), the modular double and the multiparticle q-deformed Toda chains. Commun. Math. Phys. 225, 573-609 (2002).
24. Macdonald, I. G.: A New Class of Symmetric Functions. Publ. I. R. M. A. Strasbourg, 372/S-20, Actes 20e Séminaire Lotharingien de Combinatoire, 131-171 (1988).
25. Ruijsenaars S.: The relativistic Toda systems. Commun. Math. Phys. 133, 217-247 (1990).
26. Sevostyanov A.: Regular nilpotent elements and quantum groups. Commun. Math. Phys. 204, 1-16 (1999).
27. Sevostyanov A.: Quantum deformation of Whittaker modules and Toda lattice. Duke Math. J. 105, 211-238 (2000).
28. Shintani T.: On an explicit formula for class 1 Whittaker functions on GLn over p-adic fields. Proc. Japan Acad. 52, 180-182 (1976).
29. Zhelobenko D., Shtern A.: Representations of Lie Groups. Nauka, Moscow (1983).