Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 017, 59 pages      arXiv:2505.09520      https://doi.org/10.3842/SIGMA.2026.017

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

Artem Kalmykov
Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, Quebec H3A 0B9, Canada

Received August 14, 2025, in final form February 09, 2026; Published online February 23, 2026

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.

Key words: Yangian; Toda lattice; shuffle algebra; affine Hecke algebra.

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