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? 



Representation theory of G1 versus G/Z
At least over a number field, there is a section of the map to those norms of characters, so the adele group is a product of $G^1$ with a product of some "rays" $(0,\infty)$ whose representation theory we know.


Indeed, "removing" those rays does remove some "spurious" spectral decomposition mess from the automorphic spectrum.


It is also convenient that the intersection of the adele points of the center with $G^1$, modulo the rational points of the center, is compact, so the corresponding decomposition is discrete.


Edit: what I intended by "spurious" is that the repn theory of $(0,\infty)$ is elementary, and its entanglement with automorphic repn theory is usually inessential, although potentially creating notational trouble, as opposed to serious mathematical issues.
A definite disadvantage to $G^1$ is that it is no longer a (colimit of) product(s) of local groups. (Issues about non-unique maximal compacts in p-adic reductive groups are inescapable, but the Iwahori is unique.) 


