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We establish a previously conjectured connection between $p$-adics andquantum groups. We find in Sklyanin's two parameter elliptic quantum algebraand 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 Jostfunction for the scattering of {\em first} level excitations in the$Z_n$\--Baxter model coincides with the Harish\--Chandra\--like $c$\--functionconstructed from the Macdonald polynomials associated to the root system $A_1$.The partition function of the $Z_2$\--Baxter model itself is also expressed interms of this Macdonald\--Harish\--Chandra\ $c$\--function, albeit in a lesssimple way. We relate the two parameters $q$ and $t$ of the Macdonaldpolynomials to the anisotropy and modular parameters of the Baxter model. Inparticular the $p$\--adic ``regimes'' in the Macdonald polynomials correspondto a discrete sequence of XXZ models. We also discuss the possibility of``$q$\--deforming'' Euler products.Comment: 25 page

Macdonald polynomials from Sklyanin algebras: A conceptual basis for thep-adics-quantum group connection