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Russian Mathematical Surveys, 2012, Volume 67, Issue 1, Pages 1–92
DOI: https://doi.org/10.1070/RM2012v067n01ABEH004776
(Mi rm9463)
 

This article is cited in 20 scientific papers (total in 20 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
References:
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.
Received: 14.07.2011
Bibliographic databases:
Document Type: Article
UDC: 517.986.68+517.912+519.4
MSC: Primary 22E45; Secondary 17B80, 37J35
Language: English
Original paper language: Russian
Citation: A. A. Gerasimov, D. R. Lebedev, S. V. Oblezin, “New integral representations of Whittaker functions for classical Lie groups”, Russian Math. Surveys, 67:1 (2012), 1–92
Citation in format AMSBIB
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\by A.~A.~Gerasimov, D.~R.~Lebedev, S.~V.~Oblezin
\paper New integral representations of Whittaker functions for classical Lie groups
\jour Russian Math. Surveys
\yr 2012
\vol 67
\issue 1
\pages 1--92
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  • This publication is cited in the following 20 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Успехи математических наук Russian Mathematical Surveys
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    References:99
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