MACDONALD POLYNOMIALS FROM SKLYANIN ALGEBRAS: A CONCEPTUAL BASIS FOR THE p-ADICS-QUANTUM GROUP CONNECTION

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 Zn-Baxter models. We show that in the n → ∞ limit, the Jost function for the scattering of first level excitations in the Zn-Baxter model coincides with the HarishChandra-like c-function constructed from the Macdonald polynomials associated to the root system A1. The partition function of the Z2-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.