Global $G{L}_2$ Hecke-Baxter operator

We construct a global Hecke-Baxter operator for integrable systems of arithmetic type associated with the group $G{L}_2$. This is an element of a global Hecke algebra associated with the double coset space $G{L}_2(\mathbb{Z})\mathrm{\backslash }G{L}_2(\mathbb{R})\mathrm{/}{O}_2$. Eigenvalues of the global Hecke-Baxter operator acting on the $G{L}_2$-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.