  We consider the moduli space of stable principal G-bundles over a compact
Riemann surface C of genus >1, with G a reductive algebraic group. We
explicitly construct a map F from the generic fibre of the Hitchin map to a
generalized Prym variety associated to a suitable Galois covering of C. The map
F has finite fibres. In case G=PGl(2) one can check that the generic fibre of F
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 in case the
Dynkin diagram of G does not contain components of type $B_{n}$ n>0, or when
the commutator subgroup (G,G) is simply connected the map F is injective.
