On the Arakawa lifting Part I: Eichler commutation relations

We investigate the theta correspondence of cusp forms for the dual pair $(O^{\ast }(4),\mathrm{S}p(1,1))$ 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.