Restriction of modular forms on $E_{7,3}$ to $Sp_6$

In this paper, we study the restriction of modular forms such as Ikeda type lifts and the Eisenstein series on the exceptional group of type $E_{7,3}$ to the symplectic group $Sp_6$ (rank 3). As an application, we explicitly write down the restriction when modular forms have small weight. The restriction may contain Miyawaki lifts of type I,II (CAP forms) and genuine forms whose description is compatible with Arthur's classification.