  This paper is an introduction to the use of perverse sheaves with positive
characteristic coefficients in modular representation theory. In the first
part, we survey results relating singularities in finite and affine Schubert
varieties and nilpotent cones to modular representations of reductive groups
and their Weyl groups. The second part is a brief introduction to the theory of
perverse sheaves with an emphasis on the case of positive characteristic and
integral coefficients. In the final part, we provide some explicit examples of
stalks of intersection cohomology complexes with integral or positive
characteristic coefficients in nilpotent cones, mostly in type $A$. Some of
these computations might be new.
