a Galois representation into the ${\overline{{\mathbb{Q}}}_p}$-points of the $L$-group of $G$, associated to $\pi$. But if one looks, for example, at Proposition 3.4.4 of [@cht], one sees that they can associate $p$-adic Galois representations to certain automorphic representations on certain compact unitary groups, but that the Galois representations are taking values in a group ${\mathcal G}_n$ which one can check is *not* the $L$-group of the unitary group in question (for dimension reasons, for example). In fact there are even easier examples of this phenomenon: if $\pi$ is the automorphic representation for ${\operatorname{GL}}_2/{\mathbb{Q}}$ attached to an elliptic curve over the rationals, then (if one uses the standard normalisation for $\pi$) one sees that $\pi$ has trivial central character and hence descends to an automorphic representation for ${\operatorname{PGL}}_2/{\mathbb{Q}}$ which one would surely hope to be algebraic (because it is cohomological). However, the $L$-group of ${\operatorname{PGL}}_2/{\mathbb{Q}}$ is ${\operatorname{SL}}_2$ and there is no way of twisting the Galois representation afforded by the $p$-adic Tate module of the curve so that it lands into ${\operatorname{SL}}_2({\overline{{\mathbb{Q}}}_{\ell}})$, because the cyclotomic character has no square root (consider complex conjugation). On the other hand, there do exist automorphic representation for ${\operatorname{PGL}}_2/{\mathbb{Q}}$ which have associated Galois representations into ${\operatorname{SL}}_2({\overline{{\mathbb{Q}}}_{\ell}})$; 