Let $Bun_G$ be the moduli stack of $G$-bundles on a (geometrically irreducible smooth projective) curve $C$ over a finite field $k$, where $G$ is a split reductive group over $k$. Since Weil, we know there is an adelic description $Bun_G(k)=G(k(C)) \backslash G(\mathbb A) / \prod_{x \in |C|} G(O_x)$ (see Adelic description of moduli of $G$-bundles on a curve).
We say a vector bundle $E$ on $C$ is stable, if the slope $μ(F) < μ(E)= degE / rkE$ for any non-zero proper subbundle $F$ of $E$. And a $G$-bundle $E$ on $C$ is (Ad-)stable if the associated vector bundle $Ad(E)$ is a stable vector bundle on $C$.
At least for $G= GL_n$, Is there a similar adelic description of stable $G$-bundles on $C$ inside $G(k(C)) \backslash G(\mathbb A) / \prod_{x \in |C|} G(O_x)$ ? For $G=GL_1$, there is nothing to say. Moreover, how to understand the Harder-Narasimhan filtration on $Bun_{G}$ using the adelic description?
