New integral representations of Whittaker functions for classical Lie groups

Abstract

The present paper proposes new integral representations of -Whittaker functions corresponding to an arbitrary semisimple Lie algebra with the integrand expressed in terms of matrix elements of the fundamental representations of . For the classical Lie algebras , , and a modification of this construction is proposed, providing a direct generalization of the integral representation of -Whittaker functions first introduced by Givental. The Givental representation has a recursive structure with respect to the rank of the Lie algebra , and the proposed generalization to all classical Lie algebras retains this property. It was observed elsewhere that an integral recursion operator for the -Whittaker function in the Givental representation coincides with a degeneration of the Baxter -operator for -Toda chains. In this paper -operators for the affine Lie algebras , and a twisted form of are constructed. It is then demonstrated that the relation between integral recursion operators for the generalized Givental representations and degenerate -operators remains valid for all classical Lie algebras.Bibliography: 33 titles.