  We express a Schubert expansion of the Chern-Mather class for Schubert
varieties in the even orthogonal Grassmannian via integrals involving Pfaffians
and pushforward of the small resolutions in the sense of Intersection
Cohomology (IH) constructed by Sankaran and Vanchinathan, instead of the Nash
blowup. The equivariant localization is employed to show the way of computing
the integral. As a byproduct, we present the computations. For analogy and the
completion of the method in ordinary Grassmannians, we also suggest
Kazhdan-Lusztig classes associated to Schubert varieties in the Lagrangian and
odd orthogonal Grassmannian.
