  Let $G/P$ be a complex cominuscule flag manifold. We prove a type independent
formula for the torus equivariant Mather class of a Schubert variety in $G/P$,
and for a Schubert variety pulled back via the natural projection $G/Q \to
G/P$. We apply this to find formulae for the local Euler obstructions of
Schubert varieties, and for the torus equivariant localizations of the conormal
spaces of these Schubert varieties. We conjecture positivity properties for the
local Euler obstructions and for the Schubert expansion of Mather classes. We
check the conjectures in many cases, by utilizing results of Boe and Fu about
the characteristic cycles of the intersection homology sheaves of Schubert
varieties. We also conjecture that certain `Mather polynomials' are unimodal in
general Lie type, and log concave in type A.
