Conormal varieties on the cominuscule Grassmannian-II

Let X_w be a Schubert subvariety of a cominuscule Grassmannian X , and let μ :T^*X→𝒩 be the Springer map from the cotangent bundle of X to the nilpotent cone 𝒩 . In this paper, we construct a resolution of singularities for the conormal variety T^*_XX_w of X_w in X . Further, for X the usual or symplectic Grassmannian, we compute a system of equations defining T^*_XX_w as a subvariety of the cotangent bundle T^*X set-theoretically. This also yields a system of defining equations for the corresponding orbital varieties μ (T^*_XX_w) . Inspired by the system of defining equations, we conjecture a type-independent equality, namely T^*_XX_w=π ^-1(X_w)∩μ ^-1(μ (T^*_XX_w)) . The set-theoretic version of this conjecture follows from this work and previous work for any cominuscule Grassmannian of type A, B, or C.