Global $GL_2$ Hecke-Baxter operator

We construct a global Hecke-Baxter operator for integrable systems of arithmetic type associated with the group $GL_2$. This is an element of a global Hecke algebra associated with the double coset space $GL_2(\mathbb{Z})\backslash GL_2(\mathbb{R})/O_2$. Eigenvalues of the global Hecke-Baxter operator acting on the $GL_2$-Eisenstein series are given by the corresponding global $L$-factors. This construction generalizes our previous construction of the Hecke-Baxter operators over local completions $\mathbb{R}$ and $\mathbb{Q}_p$ of the number field $\mathbb{Q}$. Presumably, zeroes of the corresponding global $L$-factors should be subjected to an arithmetic version of the Bethe ansatz equations.