  According to Laumon, an affine Springer fiber is homeomorphic to the
universal abelian covering of the compactified Jacobian of a spectral curve. We
construct equivariant deformations $f_{n}:\overline{\mathcal{P}}_{n}\to
\mathcal{B}_{n}$ of the finite abelian coverings of this compactified Jacobian,
and decompose the complex $Rf_{n,*}\mathbf{Q}_{\ell}$ as direct sum of
intersection complexes. Pass to the limit, we obtain a similar expression for
the homology of the affine Springer fibers. A quite surprising consequence is
that we can reduce the homology to its $\Lambda^{0}$-invariant subspace. As an
application, we get a sheaf-theoretic reformulation of the purity hypothesis of
Goresky, Kottwitz and MacPherson. In an attempt to solve it, we propose a
conjecture about the punctural weight of the intermediate extension of a smooth
$\ell$-adic sheaf of pure weight.
