On Sevostyanov's construction of quantum difference Toda lattices

TL;DR

This paper generalizes Sevostyanov's quantum difference Toda lattice construction for simple Lie algebras, introduces alternative Lax matrix formulations, and connects Whittaker vectors to geometric and combinatorial models.

Contribution

It extends the construction of quantum difference Toda lattices using Dynkin diagram orientations and Sevostyanov triples, with new Lax matrix formalism and geometric interpretations.

Findings

Eigenfunctions of the Toda system are generated by pairing Whittaker vectors.

Fermionic formulas for Whittaker vectors are derived.

Geometric interpretation via line bundles on Laumon moduli spaces is provided.

Abstract

We propose a natural generalization of the construction of the quantum difference Toda lattice (introduced independently by Etingof and Sevostyanov) associated to a simple Lie algebra $\mathfrak{g}$. Our construction depends on two orientations of the Dynkin diagram of $\mathfrak{g}$ and some other data (which we refer to as a pair of Sevostyanov triples). In types $A$ and $C$, we provide an alternative construction via Lax matrix formalism, generalizing the one of Kuznetsov-Tsyganov for the classical $q$-Toda. We also show that the generating function of the pairing of Whittaker vectors in the Verma modules is an eigenfunction of the corresponding modified quantum difference Toda system and derive fermionic formulas for the former in spirit of the work by Feigin-Feigin-Jimbo-Miwa-Mukhin. We give a geometric interpretation of all Whittaker vectors in type $A$ via line bundles on the Laumon moduli spaces and obtain an edge-weight path model for them, slightly generalizing the construction of Di Francesco-Kedem-Turmunkh.