Conormal varieties on the cominuscule Grassmannian-II

Abstract

Let $X_w$ be a Schubert subvariety of a cominuscule Grassmannian X, and let $\mu :T^*X\rightarrow {\mathcal {N}}$ be the Springer map from the cotangent bundle of X to the nilpotent cone ${\mathcal {N}}$. 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 $\mu (T^*_XX_w)$. Inspired by the system of defining equations, we conjecture a type-independent equality, namely $T^*_XX_w=\pi ^{-1}(X_w)\cap \mu ^{-1}(\mu (T^*_XX_w))$. The set-theoretic version of this conjecture follows from this work and previous work for any cominuscule Grassmannian of type A, B, or C.