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This thesis, done under the supervision of Filippo Viviani, is devoted to thestudy of the spectral correspondence for $G$-Higgs pairs, in the case of$G=SL(r,\mathbb{C})$, $PGL(r, \mathbb{C})$, $Sp(2r,\mathbb{C})$,$GSp(2r,\mathbb{C})$, $PSp(2r,\mathbb{C})$, over any fiber.  In the first chapter we introduce the compactified Jacobian stackparametrizing torsion-free rank-1 sheaves over a (possibly reducible, nonreduced) projective curve and we describe the spectral correspondence for Higgspairs in terms of the compactified Jacobian.  In the second chapter, we study the notion of direct and inverse image forgeneralized divisors and generalized line bundles associated to any finite,flat morphism between embeddable noetherian schemes of pure dimension 1. In thecase when we deal with projective curves over a field, we study the Norm andinverse image maps between compactified Jacobians.  In the third chapter we introduce the Prym stack associated to any finite,flat morphism between projective curves over a field, with smooth codomain. Ifthe domain is reduced with locally planar singularities and the field isalgebraically closed, we show that the usual Prym scheme is contained in thePrym stack as an open and dense subset.  In the fourth chapter, we study the fibers of the $G$-Hitchin fibration, for$G$ varying among classical Lie groups. In particular, in the case of$G=SL(r,\mathbb{C})$, $PGL(r, \mathbb{C})$, the description of any fiberinvolves the stacky fibers of the Norm map induced by the morphism from thespectral curve to the base curve. In the case of $G=Sp(2r,\mathbb{C})$,$GSp(2r,\mathbb{C})$, $PSp(2r,\mathbb{C})$, the description of any fiberinvolves the equalizer stack of two maps defined on the compactified Jacobianof the spectral curve.

The Norm map on the compactified Jacobian, the Prym stack and Spectral data for G-Higgs pairs