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

Central morphisms and Cuspidal automorphic Representations