Global GL₂ Hecke-Baxter operator

We construct a global Hecke-Baxter operator for integrable systems of arithmetic type associated with the group GL₂. This is an element of a global Hecke algebra associated with the double coset space GL₂(Z) GL₂(R)/O₂. Eigenvalues of the global Hecke-Baxter operator acting on the GL₂-Eisenstein series are given by the corresponding global L-factors. This construction generalizes our previous construction of the Hecke-Baxter operators over local completions R and Qₚ of the number field Q. Presumably, zeroes of the corresponding global L-factors should be subjected to an arithmetic version of the Bethe ansatz equations.