Remarks on metaplectic tensor products for covers of $\operatorname{GL}_r$

Shuichiro Takeda

Vol. 290 (2017), No. 1, 199–230

DOI: 10.2140/pjm.2017.290.199

Abstract

We had previously constructed a metaplectic tensor product of automorphic representations of covers of ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_r$. To be precise, let $M={\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_{r_1}\times \cdots \times {\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_{r_k}\subseteq {\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_r$ be a Levi subgroup of ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_r$, where $r=r_1+\cdots +r_k$, and $\tilde{M}$ its metaplectic preimage in the $n$-fold metaplectic cover ${\widetilde{\mathrm{GL}<!--FUNCTION\; APPLICATION-->}}_r$ of ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_r$. For automorphic representations ${\pi }_1,\dots <!--FUNCTION\; APPLICATION-->,{\pi }_k$ of ${\widetilde{\mathrm{GL}<!--FUNCTION\; APPLICATION-->}}_{r_1}(\mathbb{𝔸}),\dots <!--FUNCTION\; APPLICATION-->,{\widetilde{\mathrm{GL}<!--FUNCTION\; APPLICATION-->}}_{r_k}(\mathbb{𝔸})$, we had constructed (under certain technical assumptions, which are always satisfied when $n=2$) an automorphic representation $\pi$ of $\tilde{M}$ that can be considered as the “tensor product” of the representations ${\pi }_1,\dots <!--FUNCTION\; APPLICATION-->,{\pi }_k$.

Here we significantly simplify and generalize our previous construction without the technical assumptions mentioned above.

Keywords

automorphic representations, metaplectic groups

Mathematical Subject Classification 2010

Primary: 11F70

Milestones

Received: 17 October 2016

Revised: 10 February 2017

Accepted: 12 February 2017

Published: 7 July 2017