On Unfoldings of Some Integrals of Automorphic Functions on General Linear Groups

We use results about Fourier coefficients appearing in [T] (and some more obtained here), to obtain information for certain among the integrals of the form $$I=\int_{GL_n(\kkk)Z_n(\A)\s GL_n(\A)}\varphi(g)\phi(g)\F(E)(\tj(g))dg$$ where: $\A$ is the adele ring of a number field $\kkk$; $\varphi$ is a $GL_n(\A)$-cuspidal automorphic form; $\phi$ is a $GL_n(\A)$-automorphic function (even the trivial for some results); $E$ is a $GL_{N}(\A)$-automorphic form for a multiple $N$ of $n$; $\F(E)$ is a Fourier coefficient of $E$ for certain choices of additive functions $\F$ in a set $\BBnk[N]$ which we defined in $[T];$ $\tj$ is a diagonal embedding of $GL_n$ in $GL_N$; of course $\tj(GL_n)\in\Stab{GL_N}{\F}$; and $Z_n$ is the center of $GL_n$.