Central morphisms and Cuspidal automorphic Representations

TL;DR

This paper investigates how cuspidal automorphic representations of connected reductive groups over global fields relate under central morphisms, establishing conditions for their restriction and occurrence in each other.

Contribution

It proves a new correspondence for cuspidal automorphic representations under central morphisms between reductive groups, including both restriction and lifting results.

Findings

Restriction of cuspidal representations preserves cuspidality.

Under injective morphisms, cuspidal representations of the smaller group appear in the larger.

The results have local analogues for local fields.

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.