Conormal varieties on the cominuscule Grassmannian-II
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Mathematische Zeitschrift
Conormal varieties on the cominuscule Grassmannian-II Rahul Singh1 Received: 15 November 2018 / Accepted: 2 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract 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 TX∗ X w of X w in X . Further, for X the usual or symplectic Grassmannian, we compute a system of equations defining TX∗ X 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(TX∗ X w ). Inspired by the system of defining equations, we conjecture a type-independent equality, namely TX∗ X w = π −1 (X w ) ∩ μ−1 (μ(TX∗ 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. We work over an algebraically closed field k of good characteristic (for a definition, see [4]). Let G be a connected algebraic group whose Lie algebra g is simple. For P a conjugacy class of parabolic subgroups of G, we denote by X P the variety of parabolic subgroups of G whose conjugacy class is P . For a parabolic group P, let u P denote the Lie algebra of the unipotent radical of P. The cotangent bundle T ∗ X P of X P is then given by T ∗ X P = (P, x) ∈ X P × N x ∈ u P , where N is the variety of nilpotent elements in g. The map μ : T ∗ X P → N , given by μ(P, x) = x, is the celebrated Springer map. Let B be the conjugacy class of Borel subgroups of G. The Steinberg variety, Z P = (B, P, x) ∈ X B × X P × N x ∈ u B ∩ u P , is reducible. Each irreducible component ZwP of Z P is the conormal variety of a G-orbit closure (under the diagonal action) in X B × X P . P, In [11], Lakshmibai and Singh constructed a resolution of singularities for certain Z w P P namely where X is cominuscule (see Sect. 1.10), and the opposite Schubert variety X ,w P via the standard monomial is smooth. One also obtains a system of equations for these Z w theory for Kac-Moody groups as developed by Littelmann [9]. In this paper, we extend the aforementioned results of [11], constructing a resolution of singularities for each irreducible component ZwP ⊂ Z P of the Steinberg variety of a
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Rahul Singh [email protected] Department of Mathematics, Northeastern University, Boston, MA 02115, USA
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cominuscule Grassmannian X P . In types A and C, we also provide a system of defining equations for each component ZwP , viewed as a subvariety of X B × X P × N . This also yields a system of defining equations for certain orbital varieties, which we discuss later in this section. Before getting into the details, let us first present the irreducible components of Z P from an alternate point of view. We fix a Borel subgroup B in G, and a standard parabolic subgroup P corresponding to omitting a c
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