New integral representations of Whittaker functions for classical Lie groups

A. A. Gerasimov   ; D. R. Lebedev   ; S. V. Oblezin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 67 (2012) no. 1, pp. 1-92

Résumé

The present paper proposes new integral representations of $\mathfrak{g}$-Whittaker functions corresponding to an arbitrary semisimple Lie algebra $\mathfrak{g}$ with the integrand expressed in terms of matrix elements of the fundamental representations of $\mathfrak{g}$. For the classical Lie algebras $\mathfrak{sp}_{2\ell}$, $\mathfrak{so}_{2\ell}$, and $\mathfrak{so}_{2\ell+1}$ a modification of this construction is proposed, providing a direct generalization of the integral representation of $\mathfrak{gl}_{\ell+1}$-Whittaker functions first introduced by Givental. The Givental representation has a recursive structure with respect to the rank $\ell+1$ of the Lie algebra $\mathfrak{gl}_{\ell+1}$, and the proposed generalization to all classical Lie algebras retains this property. It was observed elsewhere that an integral recursion operator for the $\mathfrak{gl}_{\ell+1}$-Whittaker function in the Givental representation coincides with a degeneration of the Baxter $\mathscr{Q}$-operator for $\widehat{\mathfrak{gl}}_{\ell+1}$-Toda chains. In this paper $\mathscr{Q}$-operators for the affine Lie algebras $\widehat{\mathfrak{so}}_{2\ell}$, $\widehat{\mathfrak{so}}_{2\ell+1}$ and a twisted form of $\vphantom{\rule{0pt}{10pt}}\widehat{\mathfrak{gl}}_{2\ell}$ are constructed. It is then demonstrated that the relation between integral recursion operators for the generalized Givental representations and degenerate $\mathscr{Q}$-operators remains valid for all classical Lie algebras. Bibliography: 33 titles.

Détail

Keywords: Whittaker function, Toda chain, Baxter operator.

Bibliographie

[1] A. Givental, “Stationary phase integrals, quantum Toda lattices, flag manifolds and the mirror conjecture”, Topics in singularity theory, Amer. Math. Soc. Transl. Ser. 2, 180, Amer. Math. Soc., Providence, RI, 1997, 103–115 ; arXiv: alg-geom/9612001 | MR | Zbl

[2] D. Joe, B. Kim, “Equivariant mirrors and the Virasoro conjecture for flag manifolds”, Int. Math. Res. Not., 2003:15 (2003), 859–882 ; arXiv: math.AG/0210377 | DOI | MR | Zbl

[3] I. M. Gelfand, M. S. Tseitlin, “Konechnomernye predstavleniya gruppy unimodulyarnykh matrits”, Dokl. AN SSSR, 71:5 (1950), 825–828 | MR | Zbl

[4] V. V. Batyrev, “Toric degenerations of Fano varieties and constructing mirror manifolds”, The Fano Conference, Univ. Torino, Turin, 2004, 109–122 ; arXiv: alg-geom/9712034 | MR | Zbl

[5] V. V. Batyrev, I. Ciocan-Fontanine, B. Kim, D. van Straten, “Mirror symmetry and toric degenerations of partial flag manifolds”, Acta Math., 184:1 (2000), 1–39 ; arXiv: math.AG/9803108 | DOI | MR | Zbl

[6] A. Gerasimov, S. Kharchev, D. Lebedev, S. Oblezin, “On a Gauss–Givental representation of quantum Toda chain wave function”, Int. Math. Res. Not., 2006 (2006), Art. ID 96489, 23 pp. ; arXiv: math.RT/0505310 | MR | Zbl

[7] B. Kostant, “Quantization and representation theory”, Representation theory of Lie groups, Proceedings of the SRC/LMS Research Symposium (Oxford, 1977), London Math. Soc. Lecture Note Ser., 34, Cambridge University Press, Cambridge–New York, 1979, 287–316 | MR | Zbl

[8] B. Kostant, “On Whittaker vectors and representation theory”, Invent. Math., 1978, no. 2, 101–184, 48 | DOI | MR | Zbl

[9] E. T. Whittaker, G. N. Watson, A course of modern analysis, 4th ed., Cambridge Univ. Press, Cambridge, 1996, 608 pp. ; E. T. Uitteker, Dzh. N. Vatson, Kurs sovremennogo analiza, Ch. 2. Transtsendentnye funktsii, 2-e izd., Mir, M., 1963, 500 pp. | Zbl

[10] V. Pasquier, M. Gaudin, “The periodic Toda chain and a matrix generalization of the Bessel function recursion relation”, J. Phys. A, 25:20 (1992), 5243–5252 | DOI | MR | Zbl

[11] E. K. Sklyanin, “Bäcklund transformations and Baxter's $Q$-operator”, Integrable systems: from classical to quantum (Montréal, QC, 1999), CRM Proc. Lecture Notes, 26, Amer. Math. Soc., Providence, RI, 2000, 227–250 | MR | Zbl

[12] K. Rietsch, “A mirror construction for the totally nonnegative part of the Peterson variety”, Nagoya Math. J., 183 (2006), 105–142 ; arXiv: math.AG/0604170 | MR | Zbl

[13] M. A. Ol'shanetskij, M. A. Perelomov, A. G. Reyman, M. A. Semenov-Tian-Shansky, “Integrable systems. II”, Dynamical systems. VII. Integrable systems, nonholonomic dynamical systems, Encyclopaedia Math. Sci., 16, Springer-Verlag, Berlin, 1994, 83–259 | MR | MR | Zbl | Zbl

[14] G. Lusztig, “Total positivity in reductive groups”, Lie theory and geometry, Progr. Math., 123, Birkhäuser Boston, Boston, MA, 1994, 531–568 | MR | Zbl

[15] S. Fomin, A. Zelevinsky, “Double Bruhat cells and total positivity”, J. Amer. Math. Soc., 12:2 (1999), 335–380 ; arXiv: math.RT/9802056 | DOI | MR | Zbl

[16] A. Berenstein, A. Zelevinsky, “Total positivity in Schubert varieties”, Comment. Math. Helv., 72:1 (1997), 128–166 | DOI | MR | Zbl

[17] V. G. Drinfeld, V. V. Sokolov, “Lie algebras and equations of Korteweg–de Vries type”, J. Soviet Math., 30:2 (1985), 1975–2036 | DOI | MR | Zbl | Zbl

[18] A. D. Berenstein, A. V. Zelevinsky, “Tensor product multiplicities and convex polytopes in partition space”, J. Geom. Phys., 5:3 (1989), 453–472 | DOI | MR | Zbl

[19] V. Lakshmibai, “Degeneration of flag varieties to toric varieties”, C. R. Acad. Sci. Paris Sér. I Math., 321:9 (1995), 1229–1234 | MR | Zbl

[20] A. Gerasimov, D. Lebedev, S. Oblezin, On a Gauss–Givental representation for classical groups, arXiv: math.RT/0608152

[21] A. Gerasimov, D. Lebedev, S. Oblezin, Baxter $Q$-operator and Givental integral representation for $C_n$ and $D_n$, arXiv: math.RT/0609082

[22] A. Gerasimov, D. Lebedev, S. Oblezin, “Quantum Toda chains intertwined”, St. Petersburg Math. J., 22:3 (2011), 411–435 | DOI | MR | Zbl

[23] P. Etingof, “Whittaker functions on quantum groups and $q$-deformed Toda operators”, Differential topology, infinite-dimensional Lie algebras, and applications, Amer. Math. Soc. Transl. Ser. 2, 194, Amer. Math. Soc., Providence, RI, 1999, 9–25 ; arXiv: math.QA/9901053 | MR | Zbl

[24] V. G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990, xxii+400 pp. ; V. Kats, Beskonechnomernye algebry Li, Mir, M., 1993, 426 pp. | MR | Zbl | MR | Zbl

[25] T. A. Springer, “Reductive groups”, Automorphic forms, representations and $L$-functions, Proc. Sympos. Pure Math. (Oregon State Univ., Corvallis, OR, 1977), Proc. Sympos. Pure Math., 33, eds. A. Borel, W. Casselman, Amer. Math. Soc., Providence, RI, 1979, 3–27 | MR | Zbl

[26] S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure Appl. Math., 34, Academic Press, New York–London, 1978, xv+628 pp. ; S. Khelgason, Differentsialnaya geometriya i simmetricheskie prostranstva, Mir, M., 1964, 533 pp. | MR | Zbl | Zbl

[27] H. Jacquet, “Fonctions de Whittaker associées aux groupes de Chevalley”, Bull. Soc. Math. France, 95 (1967), 243–309 | MR | Zbl

[28] M. Hashizume, “Whittaker functions on semisimple Lie groups”, Hiroshima Math. J., 12:2 (1982), 259–293 | MR | Zbl

[29] A. Gerasimov, S. Kharchev, A. Morozov, M. Olshanetsky, A. Marshakov, A. Mironov, “Liouville type models in the group theory framework. I. Finite-dimensional algebras”, Internat. J. Modern Phys. A, 12:14 (1997), 2523–2583 | DOI | MR | Zbl

[30] N. Bourbaki, Éléments de mathématique. Fasc. XXXIV, Groupes et algèbres de Lie: Chapitre VI: Systèmes des racines, Hermann, Paris, 1968 ; N. Burbaki, Gruppy i algebry Li. Glavy IV–VI, Mir, M., 1972, 334 pp. | MR | Zbl

[31] R. J. Baxter, Exactly solved models in statistical mechanics, Academic Press, London, 1982, xii+486 pp. ; R. Bekster, Tochno reshaemye modeli v statisticheskoi mekhanike, Mir, M., 1985, 488 pp. | MR | Zbl

[32] A. G. Reiman, M. A. Semenov-Tyan-Shanskii, “Integriruemye sistemy”, Teoretiko-\allowbreakgruppovoi podkhod, Sovremennaya matematika, In-t kompyuternykh issledovanii, M.–Izhevsk, 2003, 352 pp.

[33] K. Rietsch, “A mirror symmetric construction of $qH^*_T(G/P)_{(q)}$”, Adv. Math., 217:6 (2008), 2401–2442 ; arXiv: math.AG/0511124v2 | DOI | MR | Zbl