
Metaplectic Tensor Products for Automorphic Representation of $\widetilde{GL}(r)$
Let $M=\operatorname{GL}_{r_1}\times\cdots\times\operatorname{GL}_{r_k}\subseteq\operatorname{GL}_r$ be a Levi
subgroup of $\operatorname{GL}_r$, where $r=r_1+\cdots+r_k$, and $\widetilde{M}$ its metaplectic preimage
in the $n$fold metaplectic cover $\widetilde{\operatorname{GL}}_r$ of $\operatorname{GL}_r$. For automorphic
representations $\pi_1,\dots,\pi_k$ of $\widetilde{\operatorname{GL}}_{r_1}(\mathbb{A}),\dots,\widetilde{\operatorname{GL}}_{r_k}(\mathbb{A})$,
we construct (under a certain
technical assumption, which is always satisfied when $n=2$) an
automorphic representation $\pi$
of $\widetilde{M}(\mathbb{A})$ 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. squareintegrable modulo center), then
the metaplectic tensor product is cuspidal (resp. squareintegrable
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.
Keywords:automorphic forms, representations of covering groups Category:11F70 