  This work is the geometric part of our proof of the weighted fundamental
lemma, which is an extension of Ng\^o Bao Ch\^au's proof of the
Langlands-Shelstad fundamental lemma.
  Ng\^o's approach is based on a study of the elliptic part of the Hichin
fibration. The total space of this fibration is the algebraic stack of Hitchin
bundles and its base space is the affine space of "characteristic polynomials".
Over the elliptic set, the Hitchin fibration is proper and the number of points
of its fibers over a finite field can be expressed in terms of orbital
integrals.
  In this paper, we study the Hitchin fibration over an open set bigger than
the elliptic set, namely the "generically regular semi-simple set". The fibers
are in general neither of finite type nor separeted. By analogy with Arthur's
truncation, we introduce the substack of $\xi$-stable Hitchin bundles. We show
that it is a Deligne-Mumford stack, smooth over the base field and proper over
the base space of "characteristic polynomials". Moreover, the number of points
of the $\xi$-stable fibers over a finite field can be expressed as a sum of
weighted orbital integrals, which appear in the Arthur-Selberg trace formula.
