On Sevostyanov's construction of quantum difference Toda lattices
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 . Our construction depends on two orientations of the Dynkin diagram of and some other data (which we refer to as a pair of Sevostyanov triples). In types and , we provide an alternative construction via Lax matrix formalism, generalizing the one of Kuznetsov-Tsyganov for the classical -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 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.
Keywords
Cite
@article{arxiv.1804.01063,
title = {On Sevostyanov's construction of quantum difference Toda lattices},
author = {Roman Gonin and Alexander Tsymbaliuk},
journal= {arXiv preprint arXiv:1804.01063},
year = {2021}
}
Comments
v1: 42 pages. v2: 43 pages, minor corrections, Conjecture 3.1 is replaced by Theorem 3.1 (whose proof is given in a separate appendix), added Remarks 3.27(b) and 3.33(b). v3: 46 pages, minor corrections, references updated, several proofs expanded