The group of automorphisms of the moduli space of principal bundles with structure group $F_4$ and $E_6$

Abstract

Let $X$ be a smooth complex projective irreducible curve of genus $g \geq 3$. Let $G$ be the simple complex exceptional Lie group $F_4$ or $E_6$ and let $M(G)$ be the moduli space of principal $G$-bundles. In this work we describe the group of automorphisms of $M(G)$. In particular, we prove that the only automorphisms of $M(F_4)$ are those induced by the automorphisms of the base curve $X$ by pull-back and that the automorphisms of $M(E_6)$ are combinations of the action of the automorphisms of $X$ by pull-back, the action of the only nontrivial outer involution of $E_6$ on $M(E_6)$ by taking the dual and the action of the third torsion of the Picard group of $X$ by tensor product. We also prove a Torelli type theorem for the moduli spaces of principal $F_4$ and $E_6$-bundles, which we use as an auxiliary result in the proof of the main theorems, but which is interesting in itself. We finally draw some conclusions about the way we can see the natural map $M(F_4) \rightarrow M(E_6)$ induced by the inclusion of groups $F_4 \hookrightarrow E_6$.