  In my 2009 paper at Inventiones, we determine the cohomology of Lubin-Tate
spaces globally using the comparison theorem of Berkovich by computing the
fibers at supersingular points of the perverse sheaf of vanishing cycle $\Psi$
of some Shimura variety of Kottwitz-Harris-Taylor type. The most difficult
argument deals with the control of maps of the spectral sequences computing the
sheaf cohomology of both Harris-Taylor perverse sheaves and those of $\Psi$. In
this paper, we bypass these difficulties using the classical theory of
representations of the mirabolic group and a simple geometric argument.
