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.
Categories: math.RT math.NT