  This article is the third one of the series \cite{Bt1}-\cite{Bt2} on
Hitchin-Frenkel-Ngo fibration and Vinberg semigroup. Ngo \cite{N} proved the
fundamental lemma for Lie algebras in equal characteristics as a consequence of
geometric stabilization. This article show the geometric stabilization in the
group case which was conjectured by Frenkel and Ngo \cite{FN}. Along the proof,
we establish an identity between orbital integrals, which is analog to
Langlands-Shelstad fundamental lemma. From this equality, we deduce a formula
for Langlands-Shelstad transfer factors which was previously only known for Lie
algebras.
