×

Baxter operator and archimedean Hecke algebra. (English) Zbl 1163.17010

Summary: We introduce Baxter integral \({\mathcal{Q}}\)-operators for finite-dimensional Lie algebras \({\mathfrak{gl}_{\ell+1}}\) and \({\mathfrak{so}_{2\ell+1}}\). Whittaker functions corresponding to these algebras are eigenfunctions of the \({\mathcal{Q}}\)-operators with the eigenvalues expressed in terms of Gamma-functions. The appearance of the Gamma-functions is one of the manifestations of an interesting connection between Mellin-Barnes and Givental integral representations of Whittaker functions, which are in a sense dual to each other. We define a dual Baxter operator and derive a family of mixed Mellin-Barnes-Givental integral representations. Givental and Mellin-Barnes integral representations are used to provide a short proof of the Friedberg-Bump and Bump conjectures for \(G = \text{GL}(\ell + 1)\) proved earlier by Stade. We also identify eigenvalues of the Baxter \({\mathcal{Q}}\)-operator acting on Whittaker functions with local Archimedean \(L\)-factors. The Baxter \({\mathcal{Q}}\)-operator introduced in this paper is then described as a particular realization of the explicitly defined universal Baxter operator in the spherical Hecke algebra \({\mathcal{H}(G(\mathbb{R}),K)}\), \(K\) being a maximal compact subgroup of \(G\). Finally we stress an analogy between \({\mathcal{Q}}\)-operators and certain elements of the non-Archimedean Hecke algebra \({\mathcal {H}(G(\mathbb{Q}_p),G(\mathbb{Z}_p))}\).

MSC:

17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
20C08 Hecke algebras and their representations
22E60 Lie algebras of Lie groups
33C80 Connections of hypergeometric functions with groups and algebras, and related topics
81R12 Groups and algebras in quantum theory and relations with integrable systems
82B23 Exactly solvable models; Bethe ansatz

Software:

GL(n)pack
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Baxter R.J.: Exactly solved models in statistical mechanics. Academic Press, London (1982) · Zbl 0538.60093
[2] Berenstein A., Zelevinsky A.: Tensor product multiplicities and convex polytopes in partition space. J. Geom. Phys. 5, 453–472 (1989) · Zbl 0712.17006
[3] Bump, D.: The Rankin-Selberg method: A survey. In: Number Theory, Trace Formulas and Discrete Groups: Symposium in Honor of Atle Selberg (Oslo, Norway, July 14–21, 1987), London: Academic Press, (1989)
[4] Casselman W., Shalika J.: The unramified principal series of p-adic groups II. The Whittaker function. Comp. Math. 41, 207–231 (1980) · Zbl 0472.22005
[5] Etingof, P.: Whittaker functions on quantum groups and q-deformed Toda operators. Amer. Math. Soc. Transl. Ser.2, 194, Providence, RI: Amer. Math. Soc., 1999, pp. 9–25 · Zbl 1157.33327
[6] Gerasimov A., Kharchev S., Lebedev D.: Representation Theory and Quantum Inverse Scattering Method: Open Toda Chain and Hyperbolic Sutherland Model. Int. Math. Res. Notices 17, 823–854 (2004) · Zbl 1084.35098
[7] Gerasimov, A., Kharchev, S., Lebedev, D., Oblezin, S.: On a Gauss-Givental representation for quantum Toda chain wave function. Int. Math. Res. Notices, Volume 2006, Article ID 96489, 23 p. · Zbl 1142.17019
[8] Gerasimov, A., Lebedev, D., Oblezin, S.: Givental representation for classical groups. http://arxiv.org/list/math.RT/0608152 , 2006 · Zbl 1142.17019
[9] Gerasimov, A., Lebedev, D., Oblezin, S.: Baxter Q-operator and Givental integral representation for C n and D n . http://arxiv.org/list/math.RT/0609082 , 2006
[10] Gerasimov, A., Lebedev, D., Oblezin, S.: New Integral Representations of Whittaker Functions for Classical Lie Groups. http://arxiv.org/abs/0705.2886 , 2007 · Zbl 1267.17007
[11] Givental, A.: Stationary Phase Integrals, Quantum Toda Lattices, Flag Manifolds and the Mirror Conjecture. In: Topics in Singularity Theory, Amer. Math. Soc. Transl. Ser., 2 180, Providence, RI: Amer. Math. Soc., 1997, pp. 103–115 · Zbl 0895.32006
[12] Goldfeld D.: Automorphic forms and L-functions for the group GL(N, R). Cambridge studies in Adv. Math. Cambridge Univ. Press, Cambridge (2006) · Zbl 1108.11039
[13] Gustafson R.A: Some q-beta and Mellin-Barnes integrals on compact Lie groups and Lie algebras. Trans. Amer. Math. Soc. 341(1), 69–119 (1994) · Zbl 0796.33012
[14] Harish-Chandra: Spherical functions on a semisimple Lie group I, II. Amer. J. Math. 80, 241–310 (1958) 553–613 · Zbl 0093.12801
[15] Hashizume M.: Whittaker functions on semi-simple Lie groups. Hiroshima Math. J. 12, 259–293 (1982) · Zbl 0524.43005
[16] Jacquet H.: Fonctions de Whittaker associées aux groupes de Chevalley. Bull. Soc. Math. France 95, 243–309 (1967) · Zbl 0155.05901
[17] Jacquet H., Piatetski-Shapiro I.I., Shalika J.: Rankin-Selberg convolutions. Amer. J. Math. 105, 367–464 (1983) · Zbl 0525.22018
[18] Jacquet H., Shalika J.: Rankin-Selberg convolution: the Archimedean theory. In: (eds) Festshrift in Honor of Piatetski-Shapiro, Part I, pp. 125–207. Amer. Math. Soc., Providence, RI (1990)
[19] Joe D., Kim B.: Equivariant mirrors and the Virasoro conjecture for flag manifolds. Int. Math. Res. Notices 15, 859–882 (2003) · Zbl 1146.14302
[20] Kac V.: Infinite-dimensional Lie algebras. Cambridge University Press, Cambridge (1990) · Zbl 0716.17022
[21] Kharchev S., Lebedev D.: Eigenfunctions of GL(N, R) Toda chain: The Mellin-Barnes representation. JETP Lett. 71, 235–238 (2000)
[22] Kharchev S., Lebedev D.: Integral representations for the eigenfunctions of quantum open and periodic Toda chains from QISM formalism. J. Phys. A 34, 2247–2258 (2001) · Zbl 0971.81172
[23] Kostant, B.: Quantization and representation theory. In: Representation theory of Lie groups. 34, London Math. Soc. Lecture Notes Series, London. Math. Soc., 1979, pp. 287–316 · Zbl 0474.58010
[24] Kostant B.: On Whittaker vectors and representation theory. Invent. Math. 48(2), 101–184 (1978) · Zbl 0405.22013
[25] Parshin A.N.: On the arithmetic of 2-dimensional schemes. I, Repartitions and residues. Russ. Math. Izv. 40, 736–773 (1976) · Zbl 0358.14012
[26] 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) · Zbl 0768.58023
[27] Reyman, A.G., Semenov-Tian-Shansky, M.A.: Integrable Systems. Group theory approach. Modern Mathematics, Moscow-Igevsk: Institute Computer Sciences, 2003 · Zbl 1030.37048
[28] Serre, J.-P.: Facteurs locaux des fonctions zêta des variétés algébraiques (définisions et conjecures). Sém. Delange-Pisot-Poitou, exp. 19, 1969/70
[29] Shalika J.A.: The multiplicity one theorem for GL n . Ann. Math. 100(1), 171–193 (1974) · Zbl 0316.12010
[30] Shintani T.: On an explicit formula for class 1 Whittaker functions on GL n over padic fields. Proc. Japan Acad. 52, 180–182 (1976) · Zbl 0387.43002
[31] Stade E.: On explicit integral formulas for \({GL(n,\mathbb R)}\) -Whittaker functions. Duke Math. J. 60(2), 313–362 (1990) · Zbl 0731.11027
[32] Stade E.: Mellin transforms of \({GL(n,\mathbb R)}\) Whittaker functions. Amer. J. Math. 123, 121–161 (2001) · Zbl 1017.11022
[33] Stade E.: Archimedean L-functions and Barnes integrals. Israel J. Math. 127, 201–219 (2002) · Zbl 1032.11020
[34] Ishii T., Stade E.: New formulas for Whittaker functions on GL(N, R). J. Funct. Anal. 244, 289–314 (2007) · Zbl 1121.22003
[35] Semenov-Tian-Shansky, M.: Quantum Toda lattices. Spectral theory and scattering. Preprint LOMI 3-84, 1984: Quantization of open Toda lattices. In ”Encyclodpaedia of Mathematical Sciences” 16. Dynamical systems VII. Berlin: Springer, 1994, pp. 226–259
[36] Vinogradov I., Takhtadzhyan L.: Theory of Eisenstein Series for the group \({SL(3,\mathbb{R})}\) and its application to a binary problem. J. Soviet. Math. 18, 293–324 (1982) · Zbl 0476.10024
[37] Weil A.: Basic Number theory. Springer, Berlin (1967) · Zbl 0176.33601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.