A decomposition theorem for the affine Springer fibers

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 $f_n:{\bar{\mathcal{P}}}_n\to {\mathcal{B}}_n$ of the finite abelian coverings of this compactified Jacobian, and decompose the complex $R{f}_{n,\ast }{\mathbf{Q}}_{\mathrm{\ell }}$ 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 ${\mathrm{\Lambda }}^0$-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 $\mathrm{\ell }$-adic sheaf of pure weight.