Metaplectic tensor products for automorphic representations of $\widetilde{GL}(r)$

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 $π_1,\dots,π_k$ of $\tilde{GL}_{r_1}(\A),\dots,\tilde{GL}_{r_k}(\A)$, we construct an automorphic representation $π$ of $\tilde{M}(\A)$ which can be considered as the "tensor product" of the representations $π_1,\dots,π_k$. This is the global analogue of the metaplectic tensor product defined by P. Mezo in the sense that locally at each place $v$, $π_v$ is equivalent to the local metaplectic tensor product of ${π_1}_v,\dots,{π_k}_v$ defined by Mezo. Then we show that if all of $π_i$ 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.

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