  Tangent spaces to Schubert varieties of type A were characterized by
Lakshmibai and Seshadri. This result was extended to the other classical types
by Lakshmibai. We give a uniform characterization of tangent spaces to Schubert
varieties in cominuscule G/P. Our results extend beyond cominuscule G/P; they
describe the tangent space to any Schubert variety in G/B at a point xB, where
x is a cominuscule Weyl group element in the sense of Peterson. Our results
also give partial information about the tangent space to any Schubert variety
at any point. Our method is to describe the tangent spaces of Kazhdan-Lusztig
varieties, and then recover results for Schubert varieties. Our proof uses a
relationship between weights of the tangent space of a variety with torus
action, and factors of the class of the variety in torus equivariant K-theory.
The proof relies on a formula for Schubert classes in equivariant K-theory due
to Graham and Willems, as well as a theorem on subword complexes due to Knutson
and Miller.
