A remark on a conjecture of Buzzard-Gee and the cohomology of Shimura varieties (Q395051)

In their recent paper \textit{K. M. Buzzard} and {T. Gee} [The conjectural connections between automorphic representations and Galois representations. To appear in Proc. LMS Durham Symposium 2011. Available at \url{NEWLINEhttp://www2.imperial.ac.uk/~tsg/}] generalize the notion of algebraicity to arbitrary connected reductive groups \(G\) over number fields \(F\). In fact, they have introduced two notions of algebraicity. An automorphic representation \(\pi\) of \(G\) is \(L\)-algebraic if one can attach to it an \(l\)-adic Galois representation into the \(L\)-group of \(G\). It is \(C\)-algebraic if \(\pi\) appears in an appropriate cohomology group. The notion of \(C\)-algebraicity generalizes Clozel's notion of algebraicity.NEWLINENEWLINEBuzzard and Gee define a canonical \(\mathbb{G}_m\)-extension \(\tilde{G}\) of \(G\) and for \(C\)-algebraic automorphic representations \(\pi\) of \(G\) they canonically construct \(L\)-algebraic automorphic representations \(\tilde{\pi}\) of \(\tilde{G}\) and hence Galois representations \(\rho _{\pi}\) into the \(L\)-group of \(\tilde{G}\) (which is by definition a \(C\)-group of \(G\)).NEWLINENEWLINEThey formulate the conjecture on association of Galois representations to \(C\)-algebraic automorphic representations. The author in this short paper compares their conjecture with the conjectural description of the cohomology of Shimura varieties due to \textit{R. Kottwitz} [Automorphic forms, Shimura varieties, and L-functions. Vol. I, Proc. Conf., Ann Arbor/MI (USA) 1988, Perspect. Math. 10, 161--209 (1990; Zbl 0743.14019)].