New integral representations of Whittaker functions for classical Lie groups. (English. Russian original) Zbl 1267.17007

Gerasimov, A. A.; Lebedev, D. R.; Oblezin, S. V.

Russ. Math. Surv. 67, No. 1, 1-92 (2012); translation from Usp. Mat. Nauk 67, No. 1, 3-96 (2012).

A remarkable integral representation for the common eigenfunctions of \(\mathfrak{gl}_{l+1}\) Toda chain Hamiltonian operators was proposed in [A. Givental, Transl., Ser. 2, Am. Math. Soc. 180(34), 103–115 (1997; Zbl 0895.32006)], see also [D. Joe and 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 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 \(\mathcal{U}(\mathfrak{g})\).
In 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).
In [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.

Reviewer: Fuhai Zhu (Tianjin)

MSC:

17B20	Simple, semisimple, reductive (super)algebras
22E45	Representations of Lie and linear algebraic groups over real fields: analytic methods
33D80	Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics

Keywords:

Whittaker function; Toda chain; Baxter operator

Citations:

Zbl 0895.32006; Zbl 1146.14302; Zbl 0405.22013; Zbl 0474.58010; Zbl 1142.17019