New Integral Representations of Whittaker Functions for Classical Lie Groups

TL;DR

This paper introduces new integral representations for Whittaker functions associated with classical Lie groups, extending Givental's recursive framework and connecting it with Baxter Q-operators for affine Lie algebras.

Contribution

It generalizes Givental's integral representation to classical Lie algebras and constructs corresponding Q-operators, preserving the recursive structure and linking to affine algebra degenerations.

Findings

Integral representations for sp(2l), so(2l), so(2l+1) Whittaker functions.

Recursive structure similar to Givental's representation is maintained.

Q-operators for affine Lie algebras are constructed and related to recursion operators.

Abstract

We propose integral representations of the Whittaker functions for the classical Lie algebras sp(2l), so(2l) and so(2l+1). These integral representations generalize the integral representation of gl(l+1)-Whittaker functions first introduced by Givental. One of the salient features of the Givental representation is its recursive structure with respect to the rank of the Lie algebra gl(l+1). The proposed generalization of the Givental representation to the classical Lie algebras retains this property. It was shown elsewhere that the integral recursion operator for gl(l+1)-Whittaker function in the Givental representation coincides with a degeneration of the Baxter Q-operator for $\hat{gl(l+1)}$-Toda chains. We construct Q-operator for affine Lie algebras $\hat{so(2l)}$, $\hat{so(2l+1)}$ and a twisted form of $\hat{gl(2l)}$. We demonstrate that the relation between recursion integral operators of the generalized Givental representation and degenerate Q-operators remains valid for all classical Lie algebras.