New integral representations of Whittaker functions for classical Lie groups (Q2892022)

A remarkable integral representation for the common eigenfunctions of \(\mathfrak{gl}_{l+1}\) Toda chain Hamiltonian operators was proposed in [\textit{A. Givental}, Transl., Ser. 2, Am. Math. Soc. 180(34), 103--115 (1997; Zbl 0895.32006)], see also [\textit{D. Joe} and \textit{B. Kim}, Int. Math. Res. Not. 2003, No. 15, 859--882 (2003; Zbl 1146.14302)]. This integral representation arises naturally in a construction of a mirror dual of Type A topological closed strings on \(\mathfrak{gl}_{l+1}\). According to \textit{B. Kostant} [Invent. Math. 48, 101--184 (1978; Zbl 0405.22013), Representation theory of Lie groups, Proc. SRC/LMS Res. Symp., Oxford 1977, Lond. Math. Soc. Lect. Note Ser. 34, 287--316 (1979; Zbl 0474.58010)], the common eigenfunctions of \(\mathfrak{g}\)-Toda chain Hamiltonian operators are given by generalizations of the classical Whittaker functions and can be expressed in terms of matrix elements of infinite-dimensional representations of the universal enveloping algebra \(\mathcal{U}(\mathfrak{g})\).NEWLINENEWLINEIn this paper the authors propose a universal construction of an integral representation of a \(\mathfrak{g}\)-Whittaker function for an arbitrary semisimple Lie algebra \(\mathfrak{g}\) with the integrands expressed in terms of the matrix elements of the fundamental representations of \(\mathfrak{g}\) (Proposition~1.1). For classical Lie algebras \(\mathfrak{so}_{2l+1}\), \(\mathfrak{sp}_{2l}\), and \(\mathfrak{so}_{2l}\), the construction is modified to obtain a generalization of the Givental construction for classical Lie algebras (Theorems~1.3, 1.6, 1.10, and 1.14).NEWLINENEWLINEIn [\textit{A. Gerasimov} et al., Int. Math. Res. Not. No. 6, 23 p. (2006; Zbl 1142.17019)], it was noted that the Givental integral representation has a recursive structure connecting the \(\mathfrak{gl}_l\)- and \(\mathfrak{gl}_{l+1}\)-Whittaker functions by simple integral transformations. The corresponding integral operator coincides with a particular degeneration of the Baxter \({\mathcal Q}\)-operator for the \(\widehat{\mathfrak{gl}}_{l+1}\)-Toda chain. In the paper under review, the authors introduce the Baxter \({\mathcal Q}\)-operators associated with the classical affine Lie algebras \(\widehat{\mathfrak{so}}_{2l}\), \(\widehat{\mathfrak{so}}_{2l+1}\) and a twisted form of \(\widehat{\mathfrak{gl}}_{2l}\) and prove that the relation between recursion integral operators of the generalized Givental representation and degenerate \({\mathcal Q}\)-operators remains valid for all classical Lie algebras.