  The purpose of this article is to set foundations for decomposition numbers
of perverse sheaves, to give some methods to calculate them in simple cases,
and to compute them concretely in two situations: for a simple (Kleinian)
surface singularity, and for the closure of the minimal non-trivial orbit in a
simple Lie algebra.
  This work has applications to modular representation theory, for Weyl groups
using the nilpotent cone of the corresponding semisimple Lie algebra, and for
reductive algebraic group schemes using the affine Grassmannian of the
Langlands dual group.
