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 $k<0$?

linebundle. Consider the case of Siegel modular forms on Sp_4: I have two weights k1,k2. I think that if k1>=k2 there's a coherent sheaf and if k1>=k2>=3 there's an etale sheaf. But I can't invert most of the coherent sheaves because I think they have rank > 1. $\endgroup$ – Kevin Buzzard May 20 '10 at 18:13