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Conormal varieties on the cominuscule Grassmannian. II. (English) Zbl 1462.14051

Summary: 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.
For Part I, see [the author and V. Lakshmibai, “Conormal varieties on the cominuscule Grassmannian”, Preprint, arXiv:1712.06737].

MSC:

14M15 Grassmannians, Schubert varieties, flag manifolds
20G05 Representation theory for linear algebraic groups
05E10 Combinatorial aspects of representation theory
14N15 Classical problems, Schubert calculus
17B08 Coadjoint orbits; nilpotent varieties
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