Let $W \subset V$ be quadratic spaces over a number field $F$.

Let $G_n=SO(V)$ and $G_m=SO(W)$ and we consider $G_m$ as a subgoup of $G_n$ via a diagonal embedding.

Let $f$ be an automorphic form of $G_n$ and $g$ an automorphic form of $G_m$.

I am wondering whether the function $h$ on $G_m$ defined by $h(k)=f(k)g(k)$ is automorphic form on $G_m$.

Except for $\mathfrak{z}$-finiteness, I verified that other properties of autumorphic forms does hold. But I am doubtful $\mathfrak{z}$-finiteness hold.

Do you have any idea on this?

Thank you very much!