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 φ ∈ K one may construct a W-Galois covering ~C of C and consider the generalized Prym variety P=Hom_W(X(T),J(~C)), where X(T) denotes the group of characters on T and J(C) the Jacobian. The connected component P₀⊂ 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(φ)_c of the Hitchin fibre to P₀ and study its degree. In case G=PGl(2) one has that the generic fibre of F:H^–1()_c→ P₀ 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.