Metaplectic Tensor Products for Automorphic Representation of (r)

Takeda, Shuichiro

Canadian journal of mathematics, Tome 68 (2016) no. 1, pp. 179-240

Résumé

Let $M\,=\,\text{G}{{\text{L}}_{{{r}_{1}}}}\,\times \,\cdots \,\times \,\text{G}{{\text{L}}_{{{r}_{k}}}}\,\subseteq \,\text{G}{{\text{L}}_{r}}$ be a Levi subgroup of $\text{G}{{\text{L}}_{r}}$ , where $r\,=\,{{r}_{1}}+\cdots +{{r}_{k}}$ , and $\widetilde{M}$ its metaplectic preimage in the $n$ -fold metaplectic cover $\widetilde{\text{G}{{\text{L}}_{r}}}$ of $\text{G}{{\text{L}}_{r}}$ . For automorphic representations ${{\pi }_{1}},\ldots ,{{\pi }_{k}}$ of ${{\widetilde{\text{GL}}}_{{{r}_{1}}}}\left( \mathbb{A} \right),\ldots ,{{\widetilde{\text{GL}}}_{{{r}_{k}}}}\left( \mathbb{A} \right)$ , we construct (under a certain technical assumption that is always satisfied when $n\,=\,2$ ) an automorphic representation $\pi $ of $\widetilde{M}\left( \mathbb{A} \right)$ that can be considered as the “tensor product” of the representations ${{\pi }_{1}},\ldots ,{{\pi }_{k}}$ . 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 ${{\text{ }\!\!\pi\!\!\text{ }}_{1,\,v}},\ldots ,{{\text{ }\!\!\pi\!\!\text{ }}_{k,\,v}}$ defined by Mezo. Then we show that if all of the ${{\text{ }\!\!\pi\!\!\text{ }}_{i}}$ 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.

Détail

DOI : 10.4153/CJM-2014-046-2

Mots-clés : 11F70, Automorphic forms, representations of covering groups