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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.
UDC:
517.986.68+
517.912+519.4
MSC: Primary
22E45; Secondary
17B80,
37J35 Received: 14.07.2011
DOI:
10.4213/rm9463
Fulltext:
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