# To which automorphic forms/rep's over a function field can we associate a Galois representation?

As far as I understand it, by the work of Lafforgue (cf. Laumon, "Cohom. of Drinfeld ... II", Thm 12.4.1) there is a Galois representation associated to an irreducible cuspidal automorphic representation $\pi$ , if $\pi$ is Steinberg at some place $\infty$.

Do we expect Galois representations also if some of these conditions do not hold?

I might be totally off here, but R. Taylor constructed in his thesis (using results from Brylinki-Labesse) Galois rep's to Hilbert modular forms by congruence methods.

Has this be studied anywhere for function fields?

Any hint, where such things are discussed, would be greatly appreciated!

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I'm definitely no expert, and I may be wrong, but you might expect L-functions associated to irreducible cuspidal automorphic representations to correspond exactly to primitive elements of the Selberg class. I don't know whether one can associate to any such primitive element a Galois representation though. –  Sylvain JULIEN Oct 23 '13 at 16:43

Let $X$ be a smooth projective geometrically irreducible curve over $\mathbb F_{q}$ a finite field and $F$ its global field. Let $\mathbf G/F$ be a split connected reductive group and $\widehat{\mathbf G}$ its Langlands dual. Let $\pi$ be an irreducible cuspidal automorphic representation of $\mathbf G(\mathbb A_{F})$. Then for all $\ell\nmid q$ there exists a $G_{F}$-representation $\sigma_{\pi,\ell}$ with values in $\widehat{\mathbf G}(\bar{\mathbb Q}_\ell)$ attached to $\pi$ in the usual sense. This result is essentially optimal, so that the answer to the question in the title seems to be "all of them".
This is a result of Laurent Lafforgue (Invent. Math. 147) when $\mathbf G=GL_{n}$ and of Vincent Lafforgue (preprint available from his webpage) in the other cases.