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 $Sp_{2g}$.