Mather classes and conormal spaces of Schubert varieties in cominuscule spaces

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 -> G/P for a parabolic subgroup Q subset of 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, and we prove this conjecture in Lie types A, B, and D. We also conjecture that certain 'Mather polynomials' are unimodal in general Lie type and log concave in type A.