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

Artem Kalmykov

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.

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

Abstract

We revisit an identification of the quantum Toda lattice for and the truncated shifted Yangian of , 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.
Paper Structure (26 sections, 60 theorems, 254 equations)

This paper contains 26 sections, 60 theorems, 254 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations for $\boldsymbol{\mathrm{GL}_N}$
  4. Quantum minors
  5. Differential operators
  6. Degenerate affine Hecke algebras and Yangians
  7. Shibukawa--Ueno $\boldsymbol{R}$-operator
  8. Degenerate affine Hecke algebras and shuffle products
  9. Shifted Yangian of $\boldsymbol{{\sl_2}}$
  10. Harish-Chandra bimodules
  11. Action of the degenerate affine Hecke algebra
  12. Parabolic restriction and difference operators
  13. Harish-Chandra bimodules for torus
  14. Parabolic restriction
  15. Vector representation
  16. ...and 11 more sections

Key Result

Theorem A

The rational Feigin--Odesskii shuffle product coincides with the one induced from degenerate affine Hecke algebras.

Theorems & Definitions (94)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Proposition 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Definition 3.1: ShibukawaUeno
  • Proposition 3.2: ShibukawaUeno
  • ...and 84 more