The spherical Hall algebra of $\overline{\operatorname{Spec}(\mathcal{O}_K)}$

TL;DR

This paper proves that for number fields with class number one, the spherical Hall algebra of the compactified spectrum of the ring of integers is isomorphic to a specific shuffle algebra linked to the Hecke L-function of the field.

Contribution

It generalizes previous results by establishing an isomorphism between the spherical Hall algebra and a Paley-Wiener shuffle algebra for certain number fields.

Findings

Established the isomorphism for class number one fields.

Connected the algebraic structure to Hecke L-functions.

Extended the framework of Hall algebras in number theory.

Abstract

We generalize a result of M. Kapranov, O. Schiffmann, and E. Vasserot by showing that, for a number field $K$ with class number one, the spherical Hall algebra of $\overline{\operatorname{Spec}(\mathcal{O}_K)}$, where $\mathcal{O}_K$ is the ring of integers of $K$, is isomorphic to the Paley-Wiener shuffle algebra associated to a Hecke $L$-function corresponding to $K$.