Conormal Varieties on the Cominuscule Grassmannian - II

Abstract

Let $X_w$ be a Schubert subvariety of a cominuscule Grassmannian $X$, and let $\mu :T^{\ast }X\to \mathcal{N}$ 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 $T_X^{\ast }{X}_w$ of $X_w$ in $X$. Further, for $X$ the usual or symplectic Grassmannian, we compute a system of equations defining $T_X^{\ast }{X}_w$ as a subvariety of the cotangent bundle $T^{\ast }X$ set-theoretically. This also yields a system of defining equations for the corresponding orbital varieties $\mu (T_X^{\ast }{X}_w)$. Inspired by the system of defining equations, we conjecture a type-independent equality, namely $T_X^{\ast }{X}_w={\pi }^{-1}(X_w)\cap {\mu }^{-1}(\mu (T_X^{\ast }{X}_w))$. The set-theoretic version of this conjecture follows from this work and previous work for any cominuscule Grassmannian of type A, B, or C.

Version 1 included similar results in type D. However, there were serious errors in the proof. All claims to results in type D have been removed in this version. We are grateful to Anna Melnikov for pointing us to these errors