$\ell$-adic Galois representations to such objects (that is, representations of the absolute Galois group of the number field over which the group is defined). The advantage of the latter approach is that it is surely the most concrete.

For the group ${\operatorname{GL}}_n$ over a number field, Clozel gave a definition of what it meant for an automorphic representation to be “algebraic”. The definition was, perhaps surprisingly, a non-trivial twist of a notion which presumably had been in the air for many years. Clozel made some conjectures predicting that algebraic automorphic representations should give rise to $n$-dimensional $\ell$-adic Galois representations (so his conjecture encapsulates Weil’s result on Hecke characters and Deligne’s theorem too). Clozel proved some cases of his conjecture, when he could switch to a unitary group and use algebraic geometry to construct the representations.

The goal of this paper is to generalise (most of) the *statement* of Clozel’s conjecture to the case where ${\operatorname{GL}}_n$ is replaced by an arbitrary connected reductive group $G$. Let us explain the first stumbling block in this programme. The naive conjecture would be of the following form: if an automorphic representation $\pi$ for $G$ is algebraic (in some reasonable sense) then there should be 