  Let F be a p-adic field and let G be a connected reductive group defined over
F. We assume p is big. Denote g the Lie algebra of G. We normalize suitably a
Fourier-transform on the space of smooth functions with compact support on
g(F), which to f associates f^. In a preceding paper, we have defined the space
FC(g(F)) of functions f such that the orbital integrals of f and of f^ are 0
for each element of g(F) which is not topologically nilpotent. These spaces are
compatible with endoscopic transfer. We assume here that G is absolutely
quasi-simple and simply connected. We define a decomposition of the space
FC(g(F)) in a direct sum of subspaces such that the endoscopic transfer becomes
(more or less) clear on each subspace. In particular, if G is quasi-split, we
describe the subspace of "stable" elements in FC(g(F)).
