Central morphisms and cuspidal automorphic representations

Labesse, Jean-Pierre; Schwermer, Joachim

doi:10.1016/j.jnt.2019.05.005

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Central morphisms and cuspidal automorphic representations

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Schwermer, Joachim
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.1016/j.jnt.2019.05.005
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arXiv:1812.03033.pdf
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Labesse, J.-P., & Schwermer, J. (2019). Central morphisms and cuspidal automorphic representations. Journal of Number Theory, 205, 170-193. doi:10.1016/j.jnt.2019.05.005.

Cite as: https://hdl.handle.net/21.11116/0000-0004-9A5B-9

Abstract

Let $F$ be a global field. Let $G$ and $H$ be two connected reductive group defined over $F$ endowed with an $F$-morphism $f: H\rightarrow G$ such that the induced morphism $H_{der}\rightarrow G_{der}$ on the derived groups is a central isogeny. Our main results yield in particular the following theorem: Given any irreducible cuspidal representation $\pi$ of $G(\mathbb A_F)$ its
restriction to $H(\mathbb A_F)$ contains a cuspidal representation $\sigma$ of
$H(\mathbb A_F)$. Conversely, assuming moreover that $f$ is an injection, any
irreducible cuspidal representation $\sigma$ of $H(\mathbb A_F)$ appears in the restriction of some cuspidal representation $\pi$ of $G(\mathbb A_F)$. This theorem has an obvious local analogue.