TITLE: Constructing coherent sheaves on Shimura varieties.
QUESTION [15 upvotes]: 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 $k<0$?

REPLY [11 votes]: As others have pointed out, the word you are looking for is automorphic vector bundle. The holomorphic automorphic forms are exactly the sections of these bundles. To define them, you start with a homogeneous bundle on the (compact) dual hermitian symmetric space. Initially, the construction of the automorphic vector bundle attached to the homogeneous bundle is analytic, but it can be made algebraic by the introduction of the standard principal bundle. It is known that the standard principle bundle has a canonical model over the reflex field, so this all works arithmetically. [The definition and proof of the existence of a canonical model of the standard principal bundle is in my 1988 Inventiones paper, after earlier work of Shimura, Harris, and others; there is a discussion of such things in Chapter III of my article in the Proceedings of the 1988 Ann Arbor conference, which are available on my website.]