So I'm coming from an algebraic geometry perspective and I'm trying to carefully piece together the story of interpreting automorphic forms as sections of vector bundles on Shimura varieties. I think I understand a fair chunk of how this works, but my shortcomings in algebra and rep theory are making it hard to see how one gets the automorphy factor from the bundles.
(For this question, I will not at all be worrying yet about compactifications.)
So let $X = G(\mathbb{R})/K$ be a Hermitian symmetric domain, and let $\hat{X} = G(\mathbb{C})/P$ be the compact dual. Of course here, $G$ is an algebraic group, $K \subset G(\mathbb{R})$ is a maximal compact subgroup, and $P \subset G(\mathbb{C})$ is a parabolic subgroup conjugate to $K$ (in $G(\mathbb{C})$!).
We have the Borel embedding $\beta : X \hookrightarrow \hat{X}$. Given a representation $(V, \sigma)$ of $K$, we get a (smooth) vector bundle $G(\mathbb{R}) \times_{K} V \to X$. Using the Weyl unitary trick, we extend this to an algebraic representation of $K_{\mathbb{C}}$, which we can then pullback to $P$, since $K_{\mathbb{C}}$ is a Levi factor of $P$. So we get an algebraic vector bundle $G(\mathbb{C}) \times_{P} V \to \hat{X}$. I believe pulling back via the Borel embedding establishes a one-to-one correspondence between vector bundles on $X$ and algebraic vector bundles on $\hat{X}$. In particular, I think this shows that the bundles $G(\mathbb{R}) \times_{K} V \to X$ are in fact holomorphic, not just smooth.
Now, for a discrete subgroup $\Gamma \subset G(\mathbb{R})$ we can form the double-quotient $\Gamma \backslash X$ which is quasi-projective by the Bailey-Borel theorem. Moreover, defining $\Gamma$ to act on the left of $G(\mathbb{R})$ and trivially on $V$, we get algebraic vector bundles
$$\Gamma \backslash G(\mathbb{R}) \times_{K} V \to \Gamma \backslash X.$$
It is my understanding, we want to identify automorphic forms as sections of these bundles (which will be algebraic sections by the above discussion, plus Bailey-Borel). However, the usual definition of automorphic forms for automorphy factor $j: \Gamma \times X \to \text{GL}(V)$ is as holomorphic functions $f: X \to V$ such that for all $\gamma \in \Gamma, x \in X$
$$f(\gamma \cdot x) = j(\gamma, x) f(x).$$
I was hoping someone could clarify how this automorphy factor emerges from the transition functions of the bundles $G(\mathbb{C}) \times_{P} V$. I believe that one should trivialize the bundle on an affine open set of $\hat{X}$ corresponding to the long element of the Weyl group (something I don't totally understand) and then study how $G(\mathbb{C})$ acts on the coordinates in the trivialization. If this can be explained in general, that would be awesome, but I'd also be interested just in the case of $SL_{2}$ or $Sp_{2g}$.
