New integral representations of Whittaker functions for classical Lie groups

The present paper proposes new integral representations of g-Whittaker functions corresponding to an arbitrary semisimple Lie algebra g with the integrand expressed in terms of matrix elements of the fundamental representations of g. For the classical Lie algebras sp sub(2)l, so sub(2)l, and so sub(2)l + 1 a modification of this construction is proposed, providing a direct generalization of the integral representation of gll + 1-Whittaker functions first introduced by Givental. The Givental representation has a recursive structure with respect to the rank l + 1 of the Lie algebra gll + 1, and the proposed generalization to all classical Lie algebras retains this property. It was observed elsewhere that an integral recursion operator for the gll + 2 - Whittaker function in the Givental representation coincides with a degeneration of the Baxter Q-operator for gll + 1-Toda chains. In this paper Q-operators for the affine Lie algebras so sub(2)l, so sub(2)l + 1 and a twisted form of gl sub(2)l are constructed. It is then demonstrated that the relation between integral recursion operators for the generalized Givental representations and degenerate Q-operators remains valid for all classical Lie algebras.