Let me first run through the setting of my question in an example I understand well; that of modular curves. If $Y_1(N)$ denotes the usual modular curve over the complexes, the quotient of the upper half plane by the congruence subgroup $\Gamma=\Gamma_1(N)$, then there are two kinds of sheaves that one often sees showing up in the theory of automorphic forms in this setting:
1) Locally constant sheaves. The ones showing up typically come from representations of $\Gamma$ of the form $Symm^{k-2}(\mathbf{C}^2)$, with $\Gamma$ acting in the obvious way on $\mathbf{C}^2$. These sheaves—call them $V_k$—are related to classical modular forms of weight $k$ via the Eichler-Shimura correspondence. They only exist for $k\geq2$ (weight 1 forms are not cohomological) and representation-theoreticically the sheaves are associated to representations of the algebraic group $SL(2)$ (the reason one starts in weight 2 rather than weight 0 is that there is a correction factor of "half the sum of the positive roots").
2) Coherent sheaves. The ones showing up here are powers $\omega^k$ of a canonical line bundle $\omega$ coming from the universal elliptic curve. The global sections of $\omega^k$ (which are bounded at the cusps) are classical modular forms of weight $k$. Although there are no classical modular forms of negative weight, the sheaf $\omega^k$ still makes sense for $k<0$ (in contrast to case 1 above). I am much vaguer about what is conceptually going on here. I have it in my mind that here $k$ is somehow a representation of the group $SO(2,\mathbf{R})$.
Now my question: what is the generalisation of this to arbitrary, say, PEL Shimura varieties? Part (1) I understand: I can consider algebraic representations of the reductive group I'm working with and for each such gadget I can make a locally constant sheaf. But Part (2) I understand less. I am guessing I can construct a big vector bundle on my moduli space coming from the abelian variety. Now, given some representation of some group or other, I can build coherent sheaves somehow, possibly by "changing the structure group" somehow. For which representations of which group does this give me a coherent sheaf on the moduli space?
Basically—what is the general yoga for supplying natural coherent sheaves on Shimura varieties, which specialises to the construction of $\omega^k$ in the modular curve case, and which explains why $\omega^k$ exists even for $k<0$?
Best Answer
If the Shimura variety is attached to the Shimura data $h:\mathbb S \to G_{/\mathbb R}$, and if as usual $K$ denotes the centralizer in $G_{/\mathbb R}$ of $h$, then the automorphic vector bundles are $G(\mathbb R)$-equivariant bundles on $X:= G(\mathbb R)/K(\mathbb R)$ attached to representations of the algebraic group $K$.
Specifically, if $V$ is the representation (over $\mathbb C$, say), then the associated vector bundle $\mathcal V$ is given by $(G(\mathbb R)\times V)/K(\mathbb R).$
Note that if $G = GL_2$ and we are in the modular curve case, then $K$ is equal to $\mathbb C^{\times}$ (thought of as an algebraic group over $\mathbb R$), its complexification is $\mathbb G_m\times \mathbb G_m$, and so there are two integral parameters describing its irreps (which are just one-dimensional in this case). One of them is the usual $k$ of $\omega^k$; the other can be chosen so as to correspond to twisting by a power of the determinant, which does not change the underlying line bundle, but changes the equivariant structure (which we don't think about so explicitly in the classical language).
Now $K$ is the Levi in a parabolic $P$ (defined over $\mathbb R$), and there is an open embedding (of complex analytic spaces) $X := G(\mathbb R)/K(\mathbb R) \hookrightarrow G(\mathbb C)/P(\mathbb C) =: D$. (If you look at Deligne's Corvalis article and unravel things, you'll see that this is how the complex structure on $X$ is defined: in the Hodge-theoretic formulation given there, $P$ is the parabolic preserving the Hodge filtration on the Hodge-structure corresponding to the base point of $X$.) The automorphic vector bundles are naturally defined on the flag variety $D$ (as $\mathcal V := (G(\mathbb C)\times V)/P(\mathbb C)$), and then restricted to $X$. (So in the modular curve case, $D$ is the projective line, and $\omega^k$ comes from $\mathcal O(k)$.)
Now the automorphic bundles, being $G$-equivariant, descend to bundles on each Shimura variety quotient $Sh(X,K_f)$ of $X \times G(\mathbb A_f)$, and have canonical models defined over the reflex field. In the PEL case, one will be able to construct these canonical models using data from the abelian varieties (think about how the abelian varieties give rise to Hodge structures with Mumford--Tate group equal to $G$ in the first place). In general, this result is due to Milne (see his answer).
Added: A colleague pointed out to me, regarding Kevin's initial question, that it can be understood much more generally. Namely, all automorphic vector bundles are locally free coherent sheaves, equivariant under the action of the finite adeles, but not all coherent sheaves necessarily have these properties! Practically nothing is known about the $K$-theory of Shimura varieties, although the question is obviously of fundamental interest. The very deep current work on cycles on Shimura varieties, from various points of view, must be the beginning of a substantial theory whose ultimate shape no one is in a position to imagine. So Kevin's question should be seen as a program for future collaboration between number theorists and algebraic geometers (at least).
Also, let me remark that I'm told that the term "automorphic vector bundle" was invented in a conversation between Michael Harris and Jim Milne in the second half of the 80s.