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

Question

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!

Answer 1

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.

That said, these results are far beyond my own expertise, so I hope real experts will chime in. Regarding your second question, there exists unpublished work on congruences for Drinfeld modular varieties, but my impression is that not much work has been done in this direction.