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On Baxter \({\mathcal Q}\) -Operators and their Arithmetic Implications

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Abstract

We consider Baxter \({\mathcal Q}\) -operators for various versions of quantum affine Toda chain. The interpretation of eigenvalues of the finite Toda chain Baxter operators as local Archimedean L-functions proposed recently is generalized to the case of affine Lie algebras. We also introduce a simple generalization of Baxter operators and local L-functions compatible with this identification. This gives a connection of the Toda chain Baxter \({\mathcal Q}\) -operators with an Archimedean version of the Polya–Hilbert operator proposed by Berry-Keating. We also elucidate the Dorey–Tateo spectral interpretation of eigenvalues of \({\mathcal Q}\) -operators. Using explicit expressions for eigenfunctions of affine/relativistic Toda chain we obtain an Archimedean analog of Casselman–Shalika–Shintani formula for Whittaker function in terms of characters.

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References

  1. Adams J., Barbash D., Vogan D.A. Jr: The Langlands classification and irreducible characters of real reductive groups. Progress in Mathematics, vol. 104. Birkhäuser, Basel (1992)

    Google Scholar 

  2. Baxter R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press, London (1982)

    MATH  Google Scholar 

  3. Beilinson, A.A.: Height pairings between algebraic cycles. In: Manin, Yu.I. (ed.) K-theory, Arithmetic and Geometry, Moscow 1984–86. Lecture Notes in Mathematics, vol. 1289, pp. 1–25. Springer, Berlin (1987). Contemp. Math. 67, 1–24 (1987)

  4. Beilinson, A., Drinfeld, V.: Opers. arXiv:math.AG/0501398

  5. Berry M.V., Keating J.P.: The Riemann zeros and eigenvalue asymptotics. SIAM Rev. 41(2), 236–266 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bump D.: Automorphic Forms and Representations. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  7. Casselman W., Shalika J.: The unramified principal series of p-adic groups II. The Whittaker function. Compositio Math. 41, 207–231 (1980)

    MathSciNet  MATH  Google Scholar 

  8. Cherednik, I.V.: Quantum groups as hidden symmetries of classic representation theory. In: Differential Geometric Methods in Theoretical Physics (Chester, 1988), pp. 47–54. World Scientific, Teaneck (1989)

  9. Cheung P., Kac V.: Quantum Calculus. Springer, Heidelberg (2001)

    Google Scholar 

  10. Connes A.: Trace formula in noncommutative geometry and the zeros of the Riemann zeta function. Sel. Math. (NS) 5, 29–106 (1999) arXiv:math.NT/9811068

    Article  MathSciNet  MATH  Google Scholar 

  11. Deligne P.: Theorie de Hodge I, in Actes du Congr‘es International des Mathematiciens, Nice, 1970, vol. 1, pp. 425–430. Gauthier-Villars, Paris (1971)

    Google Scholar 

  12. Deninger C.: On the Γ-factors attached to motives. Invent. Math. 104, 245–261 (1991)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. Deninger C.: Local L-factors of motives and regularized determinants. Invent. Math. 107, 135–150 (1992)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. Dorey P., Tateo R.: Anharmonic oscillators, the thermodynamic Bethe ansatz and nonlinear integral equations. J. Phys. A 32, L419–L425 (1999) arXiv:hep-th/9812211

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. Dorey, P., Tateo, R.: On the relation between Stokes multipliers and T-Q systems of conformal field theory. Nucl. Phys. B 563(3), 573–602(30). arXiv:hep-th/9906219

  16. Etingof, P.: Whittaker functions on quantum groups and q-deformed Toda operators. Am. Math. Soc. Transl. Ser. 2, vol. 194, pp. 9–25. American Mathematical Society, Providence (1999). arXiv:math.QA/9901053

  17. Feigin, B., Frenkel, E.: Affine Kac-Moody algebras at the critical level and Gelfand-Dikii algebras. In: Tsuchiya, A., Eguchi, T., Jimbo, M. (eds.) Infinite Analysis. Adv. Ser. in Math. Phys., vol. 16, pp. 197–215. World Scientific, Singapore (1992)

  18. Frenkel, E.: Affine algebras, Langlands duality and Bethe ansatz. In: Iagolnitzer, D. (ed.) Proceedings of the International Congress of Mathematical Physics, Paris, 1994, pp. 606–642. International Press, New York (1995). arXiv:math.QA/9506003

  19. Gerasimov, A., Kharchev, S., Lebedev, D.: Representation theory and quantum inverse scattering method: open Toda chain and hyperbolic Sutherland model. Int. Math. Res. Notes 17, 823–854 (2004). arXiv:math.QA/0204206

    Article  MathSciNet  Google Scholar 

  20. Gerasimov, A., Lebedev, D., Oblezin, S.: Baxter operator and Archimedean Hecke algebra. Commun. Math. Phys. doi:10.1007/s00220-008-0547-9. arXiv:math.RT/0706. 347

  21. Gerasimov, A., Lebedev, D., Oblezin, S.: New integral representations of Whittaker functions for classical Lie groups. arXiv:0705.2886

  22. Givental, A., Kim, B.: Quantum cohomology of flag manifolds and Toda lattices. Commun. Math. Phys. 168, 609. arXiv.org:hep-th/9312096

  23. Givental, A.: Stationary phase integrals, quantum Toda lattices, flag manifolds and the mirror conjecture. Topics in Singularity Theory. American Mathematical Society Transl. Ser., 2, vol. 180, pp. 103–115. American Mathematical Society, Providence (1997). arXiv:alg-geom/9612001

  24. Givental, A., Lee, Y.-P.: Quantum K-theory on flag manifolds, finite-difference Toda lattices and quantum groups. Invent. Math. 151, 193–219 (2003). arXiv:math/0108105

    Article  MathSciNet  MATH  Google Scholar 

  25. Gutzwiller M.: The quantum mechanical Toda lattice II. Ann. Phys. 133, 304–331 (1981)

    Article  MathSciNet  ADS  Google Scholar 

  26. Kac V.: Infinite-Dimensional Lie Algebras. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  27. Kapranov M.: Double affine Hecke algebras and 2-dimensional local fields. J. AMS 14, 239–262 (2001) arXiv:math.AG/9812021

    MathSciNet  MATH  Google Scholar 

  28. Kapranov, M.: Harmonic analysis on algebraic groups over two-dimensional local fields of equal characteristic. In: Geometry and Topology Monographs. Invitation to Higher Local Fields Part II, vol. 3, Sect. 5, pp. 255–262

  29. 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

    Article  ADS  Google Scholar 

  30. Kharchev S., Lebedev D.: Integral representations for the eigenfunctions of quantum open and periodic Toda chains from QISM formalism. J. Phys. A34, 2247–2258 (2001)

    MathSciNet  ADS  Google Scholar 

  31. Kharchev S., Lebedev D., Semenov-Tian-Shansky M.: Unitary representations of U q (sl(2, R)), the modular double and the multiparticle q-deformed Toda chains. Commun. Math. Phys. 225(3), 573–609 (2002) arXiv:hep-th/0102180

    Article  MathSciNet  ADS  MATH  Google Scholar 

  32. Khoroshkin S.M., Tolstoy V.N.: Yangian double. Lett. Math. Phys. 36, 373–402 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  33. Kostant, B.: Quantization and representation theory. In: Represenatation Theory of Lie Groups. Lecture Notes Series, vol. 34, pp. 287–316. London Mathematical Society, Oxford (1979)

  34. Kostant B.: On Whittaker vectors and representation theory. Invent. Math. 48(2), 101–184 (1978)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  35. Macdonald, I.G.: A new class of symmetric functions. Séminaire Lotharingien de Combinatoire, B20a, 41 pp (1988)

  36. Manin Yu.I.: The notion of dimension in geometry and algebra. Bull. AMS (NS) 43(2), 139–161 (2006) arXiv:math/0502016

    Article  MathSciNet  MATH  Google Scholar 

  37. Manin, Yu.: Lectures on zeta functions and motives (according to Deninger and Kurokawa), In: Columbia University Number Theory Seminar, Astérisque, vol. 228, pp. 121–164 (1995)

  38. Parshin A.N.: On the arithmetic of 2-dimensional schemes. I Repartitions and residues. Russ. Math. Izv. 40, 736–773 (1976)

    MATH  Google Scholar 

  39. Pasquier V., Gaudin M.: The periodic Toda chain and a matrix generalization of the Bessel function recursion relation. J. Phys. A 25, 5243–5252 (1992)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  40. Ruijsenaars S.: The relativistic Toda systems. Commun. Math. Phys. 133, 217–247 (1990)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  41. Serre, J.-P.: Facteurs locaux des fonctions zêta des variétés algébraiques (définisions et conjectures), Sém. Delange-Pisot-Poitou, exp. 19 (1969/1970)

  42. 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)

    Article  MathSciNet  MATH  Google Scholar 

  43. Sklyanin E.: The quantum Toda chain. Lect. Notes Phys. 226, 196–233 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  44. Stade E.: Archimedean L-functions and Barnes integrals. Isr. J. Math. 127, 201–219 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  45. Tate, J.: Number theoretic background. In: Automorphic forms and L-functions. Proceedings of Symposia in Pure Mathematics, vol. 33, part 2, pp. 3–26 (1979)

  46. Vogan D.A. Jr.: The local Langlands conjecture. In: Adams, J. et al. (eds) Representation Theory of Groups and Algebras. Contemporary Mathematics, vol. 145, pp. 305–379. American Mathematical Society, Providence (1993)

    Google Scholar 

  47. Weil A.: Basic Number theory. Springer, Heidelberg (1967)

    MATH  Google Scholar 

  48. Whittaker E.T., Watson G.N.: A Course of Modern Analysis. Cambridge University Press, London (1962)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institute for Theoretical and Experimental Physics, 117259, Moscow, Russia

    Anton Gerasimov

  2. School of Mathematics, Trinity College, Dublin 2, Ireland

    Anton Gerasimov

  3. Hamilton Mathematics Institute, TCD, Dublin 2, Ireland

    Anton Gerasimov

  4. Institute for Theoretical and Experimental Physics, 117259, Moscow, Russia

    Dimitri Lebedev & Sergey Oblezin

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  1. Anton Gerasimov
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  2. Dimitri Lebedev
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  3. Sergey Oblezin
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Correspondence to Dimitri Lebedev.

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Gerasimov, A., Lebedev, D. & Oblezin, S. On Baxter \({\mathcal Q}\) -Operators and their Arithmetic Implications. Lett Math Phys 88, 3–30 (2009). https://doi.org/10.1007/s11005-008-0285-0

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