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

##### 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
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