Macdonald Polynomials from Sklyanin Algebras: A Conceptual Basis for the $p$-Adics-Quantum Group Connection

TL;DR

This paper reveals a deep connection between $p$-adic analysis, quantum groups, and Macdonald polynomials through Sklyanin algebras, providing a conceptual framework linking integrable models, symmetric spaces, and special functions.

Contribution

It establishes a conceptual basis for the relation between Macdonald polynomials and $p$-adic quantum groups via Sklyanin algebras, connecting integrable models and symmetric spaces.

Findings

Jost function matches Macdonald-based Harish-Chandra $c$-function in the $n o fty$ limit.

Partition function of the $Z_2$-Baxter model expressed via Macdonald-Harish-Chandra $c$-function.

$q$ and $t$ parameters relate to anisotropy and modular parameters of the Baxter model.

Abstract

We establish a previously conjectured connection between $p$-adics and quantum groups. We find in Sklyanin's two parameter elliptic quantum algebra and its generalizations, the conceptual basis for the Macdonald polynomials, which ``interpolate'' between the zonal spherical functions of related real and $p$\--adic symmetric spaces. The elliptic quantum algebras underlie the $Z_n$\--Baxter models. We show that in the $n \air \infty$ limit, the Jost function for the scattering of {\em first} level excitations in the $Z_n$\--Baxter model coincides with the Harish\--Chandra\--like $c$\--function constructed from the Macdonald polynomials associated to the root system $A_1$. The partition function of the $Z_2$\--Baxter model itself is also expressed in terms of this Macdonald\--Harish\--Chandra\ $c$\--function, albeit in a less simple way. We relate the two parameters $q$ and $t$ of the Macdonald polynomials to the anisotropy and modular parameters of the Baxter model. In particular the $p$\--adic ``regimes'' in the Macdonald polynomials correspond to a discrete sequence of XXZ models. We also discuss the possibility of ``$q$\--deforming'' Euler products.