Let $G$ be an locally compact group $G$, then every irreucible representations $\pi$ is isomorphic to $\omega_{\pi} \otimes \pi'$, where $\omega_{\pi}$ is the central character of $\pi$ and $\pi'$ an irreducible representation with trivial central character, hence it is sufficient to classify, construct, analyse ... only the irreducible representation of $G/Z$.

Let $G$ now be a reductive group over a global field, why is it sufficient to study the representation theory of $G(\mathbb{A})^1$, where $G(\mathbb{A})^1$ is the intersection of all kernels of $x \mapsto |\chi(x) |_{\mathbb{A}}$ for $\chi$ a rational character $\chi$, in order to understand all automorphic representations?

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