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

Abstract

Let $M=G{L}_{r_1}\times \cdots \times G{L}_{r_k}\subseteq G{L}_r$ be a Levi subgroup of $G{L}_r$, where $r=r_1+\cdots +r_k$, and $\tilde{M}$ its metaplectic preimage in the metaplectic cover ${\widetilde{GL}}_r$ of $G{L}_r$. 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.

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