Non-vanishing of Miyawaki type lift

Miyawaki type lifts are kinds of Langlands functorial lifts and a special case was first conjectured by Miyawaki and proved by Ikeda for Siegel cusp forms. Since then, such a lift for Hermitian modular forms was constructed by Atobe and Kojima , and for half-integral weight Siegel cusp forms by Hayashida, and we constructed Miyawaki type lift for ${\rm GSpin}(2,10)$. Recently Ikeda and Yamana generalized Ikeda type construction and accordingly did Miyawaki type lift for Hilbert cusp forms in a remarkable way. In all these works, a construction of Miyawaki type lift takes two steps as follows: First, construct Ikeda type lift on a bigger group from an elliptic cusp form, and then define a certain integral on a block diagonal element which is an analogue of pull-back formula studied by Garrett for Siegel Eisenstein series. If the integral is non-vanishing, we show that it is a Hecke eigen cusp form, and it is the Miyawaki type lift. The question of non-vanishing of the integral was left open. In this paper, we show the non-vanishing for certain special cases.