The ``Fundamental Lemma'' is a series of combinatorial identities which have been discovered by Langlands and which have been precisely formulated by Langlands and Shelstad. Over a non archimedian local field of equal characteristics, the orbital integrals which enter in the statement of the ``Fundamental Lemma'' for unitary groups are directly related to the affine Springer fibers for the general linear groups and it is not difficult to formulate a geometric conjecture which implies the ``Fundamental Lemma''. In this talk, I would like to explain a natural link between those affine Springer fibers and some compactified Jacobians of singular curves. Moreover, using this link, I would like to show that the above geometric conjecture follows from the purity conjecture for the cohomology of the affine Springer fibers of Goresky, Kottwitz and MacPherson.