On the Arakawa lifting Part I: Eichler commutation relations

TL;DR

This paper studies the Arakawa lifting, a theta correspondence for certain automorphic forms, establishing Hecke operator relations at all places and providing an adelic reformulation with explicit commutation relations.

Contribution

It introduces the adelic reformulation of Arakawa lifting and derives Eichler commutation relations for Hecke operators at all non-Archimedean places.

Findings

Hecke operators satisfy specific commutation relations under Arakawa lifting.

Eichler commutation relations are established at all non-Archimedean places.

An Archimedean analogue using reproducing kernels is provided.

Abstract

We investigate the theta correspondence of cusp forms for the dual pair $(O^*(4),{\mathrm Sp}(1,1))$ originally introduced by Tsuneo Arakawa in the non-adelic setting. We call this Arakawa lifting. In this paper, reformulating the theta correspondence in the adelic setting, we provide commutation relations of Hecke operators satisfied by Arakawa lifting at all non-Archimedean places, which is referred to as Eichler commutation relations for classical modular forms. Their Archimedean analogue is also given in terms of reproducing kernels.