  Let $M=GL_{r_1}\times\cdots\times GL_{r_k}\subseteq GL_r$ be a Levi subgroup
of $GL_r$, where $r=r_1+\cdots+r_k$, and $\tilde{M}$ its metaplectic preimage
in the metaplectic cover $\tilde{GL}_r$ of $GL_r$. For automorphic
representations $\pi_1,\dots,\pi_k$ of
$\tilde{GL}_{r_1}(\A),\dots,\tilde{GL}_{r_k}(\A)$, we construct an automorphic
representation $\pi$ of $\tilde{M}(\A)$ which can be considered as the "tensor
product" of the representations $\pi_1,\dots,\pi_k$. This is the global
analogue of the metaplectic tensor product defined by P. Mezo in the sense that
locally at each place $v$, $\pi_v$ is equivalent to the local metaplectic
tensor product of ${\pi_1}_v,\dots,{\pi_k}_v$ defined by Mezo. Then we show
that if all of $\pi_i$ are cuspidal (resp. square-integrable modulo center),
then the metaplectic tensor product is cuspidal (resp. square-integrable modulo
center). We also show that (both locally and globally) the metaplectic tensor
product behaves in the expected way under the action of a Weyl group element,
and show the compatibility with parabolic inductions.
