  In the geometric situation of some simple unitary Shimura varieties studied
by Harris and Taylor, I have built two filtrations of the perverse sheaf of
vanishing cycles. The graduate of the first are the $p$-intermediate extension
of some local Harris-Taylor's local systems, while for the second, obtained by
duality, they are the $p+$-intermediate extensions. In this work, we describe
the difference between these $p$ and $p+$ intermediate extension. Precisely, we
show, in the case where the local system is associated to an irreducible
cuspidal representation whose reduction modulo $l$ is supercuspidal, that the
two intermediate extensions are the same. Otherwise, if the reduction modulo
$l$ is just cuspidal, we describe the $l$-torsion of their difference.
