On the length of global integrals for $GL_n$

In this paper we prove Conjecture \ref{conj1} for a set of representations of the group $GL_n({\bf A})$. This Conjecture is stated in complete generality as Conjecture 1 in \cite{G2}, and here we prove it for various cases. See Conjecture \ref{conj2} below. First we prove it in the case when the length of the integral is four, and then we discuss the general case.

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