Stable vector bundles in Weil's parametrization

Question

Let $C$ be a smooth projective algebraic curve. Isomorphism classes of vector bundles on $C$ are in bijection with $GL_n(F) \backslash GL_n (\mathbb{A}) / GL_n(\mathcal{O})$, I think (one trivalizes over every formal disk, and patches...). Some vector bundles are called stable, semi-stable, etc. (one has more refined things like Harder-Narisimhan). My question is; how are these notions spelled out in the above parametrization?

Answer 1

I'm not sure if this is actually "spelled out", but here is a characterization of unstable vector bundles.

First recall the degree map $GL_1(\mathbb A) \to \mathbb Z$ which sends an idele to the sum over points of the degree of that point times the valuation of the local field element at that point. (Zeros are positive and poles are negative)

The unstable vector bundles are the image under the projection $GL_n(\mathbb A) \to GL_n(F)\backslash GL_n(\mathbb A)/GL_n(\mathcal O)$ of the union over $k \in \{1,\dots,n-1\}$ of the locus of block-upper-triangular matrices

$$ \begin{pmatrix} A & B \\ 0 & C \end{pmatrix}$$

where $A$ is a $k\times k$ matrix, $B$ is a $k \times n-k$ matrix, and $C$ is an $n-k \times n-k$ matrix, such that

$$ \frac{ \deg(\det(A))}{k}< \frac{\deg(\det(C))}{n-k}$$

This follows pretty straightforwardly from the usual definition of semistability when you see that block decompositions like this correspond to extensions of vector bundles, the determinant map to determinants of vector bundles, and the degree map to the degree of a line bundle. The most difficult thing is to get the signs right, which I think I did but it might be the opposite.