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

TL;DR

This paper generalizes a result about the induction map from B-bundles to semistable G-bundles over elliptic curves, showing it is small with a Galois group related to the Weyl group for all degrees.

Contribution

It extends the previous work to all connected components of the moduli stack for any reductive group G, identifying unique parabolic reductions and the structure of the induction map.

Findings

Induction map is small with Galois group as the relative Weyl group.

Existence of unique parabolic subgroup for each degree.

Provides new examples of automorphic sheaves as constituents of Eisenstein sheaves.

Abstract

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. This provides new examples of simple automorphic sheaves which are constituents of Eisenstein sheaves for the trivial local system.