An elememtary approach to the abelianization of the Hitchin system for arbitrary reductive groups

Abstract

We consider the moduli space $M$ of stable principal $G$-bundles over a compact Riemann surface $C$ of genus $g ⩾ 2, G$ being any reductive algebraic group and give an explicit description of the generic fibre of the Hitchin map $H:T^*M→ K$. If $T ⊂ G$ is a fixed maximal torus with Weyl group $W$, for each given generic element $&phis; ∈ K$ one may construct a $W$-Galois covering $&Ctilde;$ of $C$ and consider the generalized Prym variety $P=Hom_W(X(T),J(&Ctilde;))$, where $X(T)$ denotes the group of characters on $T$ and $J(&Ctilde;)$ the Jacobian. The connected component $P_0 ⊂ P$ which contains the trivial element is an abelian variety. In the present paper we use the classical theory of representations of finite groups to compute dim $P = dim M$. Next, by means of mostly elementary techniques, we explicitly construct a finite map $F$ from each connected component $H^-1(&phis;)_c$ of the Hitchin fibre to $P_0$ and study its degree. In case $G=PGl(2)$ one has that the generic fibre of $F:H^-1$()$_c → P_0$ is a principal homogeneous space with respect to a product of $(2d-2)$ copies of $Z/2Z$ where $d$ is the degree of the canonical bundle over $C$. However if the Dynkin diagram of $G$ does not contain components of type $B_l, l⩾ 1$ or when the commutator subgroup $(G,G)$ is simply connected the map $F$ is injective.