TITLE: Is there a canonical notion of "mod-l automorphic representation"?
QUESTION [10 upvotes]: As the title says.
In particular, I am interested in the story for a general reductive group $G$, say defined over $\mathbb{Q}$.  I can imagine that mod-$\ell$ (algebraic) automorphic representation should correspond to conjugacy classes of homomorphisms from the motivic Galois group into $\widehat{G}(\overline{\mathbb{F}_\ell})$, but this does not seem like a very workable definition.  Is there a more pragmatic approach?

REPLY [5 votes]: If $G$ is a connected, linear, reductive group over $\mathbb Q$, let $q_0$ be the dimension so denoted in Borel--Wallach, i.e. $q_0 = (d - l_0)/2,$ where $d = \text{dim } G_{\infty} - 
\text{ dim } A_{\infty}K_{\infty}$ and $l_0 = \text{rank } G_{\infty} - 
\text{ rank } A_{\infty}K_{\infty},$  where $G_{\infty} = G(\mathbb R)$, $K_{\infty}$ is a maximal compact subgroup of $G_{\infty}$,  and $A_{\infty} = A(\mathbb R)$ with $A$ being the maximal $\mathbb Q$-split torus in the centre of $G$.  (So $q_0$ is the first interesting dimension of homology in the locally symmetric arithmetic quotients attached to $G$.)
Then the inductive limit of the cohomology groups $H^{q_0}(G(\mathbb Q)\backslash
G(\mathbb A)/A_{\infty}K_{\infty}K_f,\mathbb F_{\ell})$, as $K_f$ runs over all compact open subgroups of $G(\mathbb A_f)$, is an admissible smooth representation of $G(\mathbb G_f)$,
which is (I believe) a reasonable substitute for a space of mod $\ell$ automorphic forms.  
In the case when $G$ satifies the conditions of Gross's "Algebraic modular forms" paper,
$q_0 = 0$, the locally symmetric quotients are just finite sets, and this is precisely the space of mod $\ell$ algebraic modular forms.
This doesn't deal with Kevin's problems; this space will (conjecturally) only contain
automorphic forms whose systems of Hecke eigenvalues correspond to odd Galois representations.  Nevertheless, it is a space, and one can (and does) try to work with it.