made by Clozel–Harris–Taylor in the unitary group case.

In this paper, we explain the phenomena above in more detail. In particular we formulate a conjecture associating $p$-adic Galois representations to $L$-algebraic automorphic representations for an arbitrary connected reductive group over a number field, which appears to essentially include all known theorems and conjectures of this form currently in the literature. We initially imagined that such a conjecture was already “known to the experts”. However, our experience has been that this is not the case; in fact, it seems that the issues that arise when comparing the definitions of $L$-algebraic and $C$-algebraic representations were a known problem, with no clear solution (earlier attempts to deal with this issue have been by means of redefining the local Langlands correspondence and the Satake isomorphism via a twist, as in [@MR1729443]; however this trick only works for certain groups). In one interesting example in the literature where Galois representations are attached to certain cohomological automorphic representations—the constructions of [@cht]—the Galois representations take values in a group $\mathcal{G}_n$, whose construction seemed to us to be one whose main motivation was that it was the group that worked, rather than the group that came from a 