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.