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 Uspekhi Mat. Nauk, 2012, Volume 67, Issue 1(403), Pages 3–96 (Mi umn9463)

New integral representations of Whittaker functions for classical Lie groups

A. A. Gerasimovab, D. R. Lebedeva, S. V. Oblezina

a Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center), Moscow
b The Hamilton Mathematics Institute, Trinity College Dublin, Ireland

Abstract: 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.

Keywords: Whittaker function, Toda chain, Baxter operator.

DOI: https://doi.org/10.4213/rm9463

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English version:
Russian Mathematical Surveys, 2012, 67:1, 1–92

Bibliographic databases:

UDC: 517.986.68+517.912+519.4
MSC: Primary 22E45; Secondary 17B80, 37J35

Citation: A. A. Gerasimov, D. R. Lebedev, S. V. Oblezin, “New integral representations of Whittaker functions for classical Lie groups”, Uspekhi Mat. Nauk, 67:1(403) (2012), 3–96; Russian Math. Surveys, 67:1 (2012), 1–92

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/umn9463
• https://doi.org/10.4213/rm9463
• http://mi.mathnet.ru/eng/umn/v67/i1/p3

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. Gerasimov, D. Lebedev, S. Oblezin, “On a classical limit of $q$-deformed Whittaker functions”, Lett. Math. Phys., 100:3 (2012), 279–290
2. N. O'Connell, “Directed polymers and the quantum Toda lattice”, Ann. Probab., 40:2 (2012), 437–458
3. A. A. Gerasimov, D. R. Lebedev, “Representation theory over tropical semifield and Langlands duality”, Comm. Math. Phys., 320:2 (2013), 301–346
4. I. Cherednik, Ma Xiaoguang, “Spherical and Whittaker functions via DAHA II”, Selecta Math. (N.S.), 19:3 (2013), 819–864
5. T. Ishii, T. Oda, “Calculus of principal series Whittaker functions on $SL(n,\mathbf{R})$”, J. Funct. Anal., 266:3 (2014), 1286–1372
6. Goncharov A., Shen L., “Geometry of Canonical Bases and Mirror Symmetry”, 202, no. 2, 2015, 487–633
7. Bisi E., Zygouras N., “Point-to-Line Polymers and Orthogonal Whittaker Functions”, Trans. Am. Math. Soc., 371:12 (2019), 8339–8379
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