DOI: 10.1017/nmj.2026.10130 ISSN: 0027-7630
On the Arakawa lifting Part I: Eichler commutation relations
Atsushi Murase, Hiro-aki Narita Abstract
We investigate the theta correspondence of cusp forms for the dual pair
(
O
∗
(
4
)
,
Sp
(
1
,
1
)
)
$(O^*(4),\mathrm {Sp}(1,1))$
left parenthesis upper O Superscript asterisk Baseline left parenthesis 4 right parenthesis comma upper S p left parenthesis 1 comma 1 right parenthesis right parenthesis
originally introduced by Tsuneo Arakawa in the non-adelic setting. We call this Arakawa lifting. In this article, reformulating the theta correspondence in the adelic setting, we provide commutation relations of Hecke operators satisfied by the Arakawa lifting at all non-archimedean places, which is referred to as Eichler commutation relations for classical modular forms. Their archimedean analog is also given in terms of reproducing kernels.