Shuffle Products, Degenerate Affine Hecke Algebras, and Quantum Toda Lattice

TL;DR

This paper provides an algebraic reinterpretation of the quantum Toda lattice and related structures, connecting degenerate affine Hecke algebras with shuffle products and offering new insights into their algebraic relationships.

Contribution

It introduces a purely algebraic approach to understanding the quantum Toda lattice and its relation to shifted Yangians, bypassing topological methods.

Findings

Algebraic interpretation of the Gerasimov-Kharchev-Lebedev-Oblezin homomorphism

Relation between degenerate affine Hecke algebras and Feigin-Odesskii shuffle products

Presentation of shuffle products via mirabolic Kostant-Whittaker reduction

Abstract

We revisit an identification of the quantum Toda lattice for $\mathrm{GL}_N$ and the truncated shifted Yangian of $\mathfrak{sl}_2$, as well as related constructions, from a purely algebraic point of view, bypassing the topological medium of the homology of the affine Grassmannian. For instance, we interpret the Gerasimov-Kharchev-Lebedev-Oblezin homomorphism into the algebra of difference operators via a finite analog of the Miura transform. This algebraic identification is deduced by relating degenerate affine Hecke algebras to the simplest example of a rational Feigin-Odesskii shuffle product. As a bonus, we obtain a presentation of the latter via a mirabolic version of the Kostant-Whittaker reduction.