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

This article is cited in 8 scientific papers (total in 8 papers)

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

Full text: PDF file (1238 kB)
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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
Received: 14.07.2011

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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    Citing articles on Google Scholar: Russian citations, English citations
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    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  crossref  mathscinet  zmath  isi  elib  scopus
    2. N. O'Connell, “Directed polymers and the quantum Toda lattice”, Ann. Probab., 40:2 (2012), 437–458  crossref  mathscinet  zmath  isi  scopus
    3. A. A. Gerasimov, D. R. Lebedev, “Representation theory over tropical semifield and Langlands duality”, Comm. Math. Phys., 320:2 (2013), 301–346  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    4. I. Cherednik, Ma Xiaoguang, “Spherical and Whittaker functions via DAHA II”, Selecta Math. (N.S.), 19:3 (2013), 819–864  crossref  mathscinet  zmath  isi  scopus
    5. T. Ishii, T. Oda, “Calculus of principal series Whittaker functions on $SL(n,\mathbf{R})$”, J. Funct. Anal., 266:3 (2014), 1286–1372  crossref  mathscinet  zmath  isi  elib  scopus
    6. Goncharov A., Shen L., “Geometry of Canonical Bases and Mirror Symmetry”, 202, no. 2, 2015, 487–633  crossref  mathscinet  zmath  isi  scopus
    7. Bisi E., Zygouras N., “Point-to-Line Polymers and Orthogonal Whittaker Functions”, Trans. Am. Math. Soc., 371:12 (2019), 8339–8379  crossref  isi
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