group over a number field.

On the other hand, one must not leave $C$-algebraic representations behind. For example, for certain unitary groups of even rank over the rationals, all automorphic representations are $C$-algebraic and none are $L$-algebraic at all! It would be a shame to have no conjecture at all in these cases. We show in section \[section:twisting\] that given a $C$-algebraic automorphic representation for a group $G$, it can be lifted to a $C$-algebraic representation for a certain covering group $\widetilde{G}$ (a canonical central extension of $G$ by $GL_1$) where there is enough space to twist $C$-algebraic representations into $L$-algebraic ones. After making such a twisting one would then conjecturally expect an associated Galois representation into the $L$-group not of $G$ but of $\widetilde{G}$. We define the $C$-group ${}^CG$ of $G$ to be ${}^L\widetilde{G}$. For example, if $\pi$ is the automorphic representation for the group ${\operatorname{PGL}}_2/{\mathbb{Q}}$ attached to an elliptic curve over ${\mathbb{Q}}$, we can verify that $\pi$ is $C$-algebraic, and that $\widetilde{G}={\operatorname{GL}}_2/{\mathbb{Q}}$ in this case, and hence one would expect a Galois representation into ${\operatorname{GL}}_2({\overline{{\mathbb{Q}}}_{\ell}})$ associated to $\pi$, which is given by the Tate module of the curve. We also verify the compatibility of the construction with that 