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

Abstract

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 $$I=\int_{GL_n(\kkk)Z_n(\A)\s GL_n(\A)}φ(g)ϕ(g)\F(E)(\tj(g))dg$$ where: $\A$ is the adele ring of a number field $\kkk$; $\phi$ is a $GL_n(\A)$-cuspidal automorphic form; $\varphi$ 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 $G{L}_n$ in $G{L}_N$; of course $\tj(GL_n)\in\Stab{GL_N}{\F}$; and $Z_n$ is the center of $G{L}_n$.

References in corpus (3)

• On Fourier Coefficients of GL(n)-Automorphic Functions over Number Fields
• On the length of global integrals for $G{L}_n$
• A Note on a Fourier Expansion of Automorphic Functions that Nontrivially Involves the Euclidean Algorithm