A path model for Whittaker vectors

TL;DR

This paper develops weighted path models to explicitly compute Whittaker vectors and functions across various Lie algebra types, revealing their connections to quantum Toda equations and $q$-difference equations.

Contribution

It introduces a unified path model approach for Whittaker vectors and functions for all finite and affine Lie algebras and quantum groups, providing new series expressions and differential equations.

Findings

Series expressions for Whittaker functions

Derivation of quantum Toda equations

Identification of $q$-difference equations

Abstract

In this paper we construct weighted path models to compute Whittaker vectors in the completion of Verma modules, as well as Whittaker functions of fundamental type, for all finite-dimensional simple Lie algebras, affine Lie algebras, and the quantum algebra $U_q(\mathfrak{sl}_{r+1})$. This leads to series expressions for the Whittaker functions. We show how this construction leads directly to the quantum Toda equations satisfied by these functions, and to the $q$-difference equations in the quantum case. We investigate the critical limit of affine Whittaker functions computed in this way.