for example one can easily build them from automorphic representations on ${\operatorname{GL}}_2/{\mathbb{Q}}$ constructed via the Langlands-Tunnell theorem applied to a continuous even irreducible 2-dimensional representation of ${\operatorname{Gal}}({\overline{{\mathbb{Q}}}}/{\mathbb{Q}})$ into ${\operatorname{SL}}_2({\mathbb{C}})$ with solvable image. What is going on?

Our proposed solution is the following. For a general connected reductive group $G$, we believe that there are *two* reasonable notions of “algebraic”. For ${\operatorname{GL}}_n$ these notions differ by a twist (and this explains why this twist appears in Clozel’s work). For some groups the notions coincide. But for some others—for example ${\operatorname{PGL}}_2$— the notions are disjoint. The two definitions “differ by half the sum of the positive roots”. We call the two notions $C$-algebraic and $L$-algebraic. It turns out that cohomological automorphic representations are $C$-algebraic (hence the $C$), and that given an $L$-algebraic automorphic representation one might expect an associated Galois representation into the $L$-group (hence the $L$). Clozel twists $C$-algebraic representations into $L$-algebraic ones in his paper, and hence conjectures that there should be Galois representations attached to $C$-algebraic representations for ${\operatorname{GL}}_n$, but this trick is not possible in general. In this paper we explicitly conjecture the existence of $p$-adic Galois representations associated to $L$-algebraic automorphic representations for a general connected reductive 