Central morphisms and Cuspidal automorphic Representations

Let $F$ be a global field. Let $G$ and $H$ be two connected reductive group defined over $F$ endowed with an $F$-morphism $f:H\to G$ such that the induced morphism $H_{der}\to G_{der}$ 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.