Abstract
We establish a previously conjectured connection betweenp-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 andp-adic symmetric spaces. The elliptic quantum algebras underlie theZ n -Baxter models. We show that in then→∞ limit, the Jost function for the scattering offirst level excitations in the 1+1 dimensional field theory model associated to theZ n -Baxter model coincides with the Harish-Chandra-likec-function constructed from the Macdonald polynomials associated to the root systemA 1. The partition function of theZ 2-Baxter model itself is also expressed in terms of this Macdonald-Harish-Chandrac-function, albeit in a less simple way. We relate the two parametersq andt of the Macdonald polynomials to the anisotropy and modular parameters of the Baxter model. In particular thep-adic “regimes” in the Macdonald polynomials correspond to a discrete sequence of XXZ models. We also discuss the possibility of “q-deforming” Euler products.
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Communicated by N.Yu. Reshetikhin
Work supported in part by the NSF: PHY-9000386
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Freund, P.G.O., Zabrodin, A.V. Macdonald polynomials from Sklyanin algebras: A conceptual basis for thep-adics-quantum group connection. Commun.Math. Phys. 147, 277–294 (1992). https://doi.org/10.1007/BF02096588
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DOI: https://doi.org/10.1007/BF02096588