On the stack of semistable $G$-bundles over an elliptic curve

Understand

In a recent paper Ben-Zvi and Nadler proved that the induction map from $B$-bundles of degree 0 to semistable $G$-bundles of degree 0 over an elliptic curve is a small map with Galois group isomorphic to the Weyl group of $G$.

• We generalize their result to all connected components of $\Bun_G$ for an arbitrary reductive group $G$.
• We prove that for every degree (i.e.
• topological type) there exists a unique parabolic subgroup such that any semistable $G$\nobreakdash-bundle of this degree has a reduction to it and moreover the induction map is small with Galois group the relative Weyl group of the Levi.