Vanishing cycles on Grassmannians
type: Item
instance of: scholarly article
title: Vanishing cycles on Grassmannians
zbMATH Open document ID: 0727.14027
DOI: 10.1215/S0012-7094-90-06128-9
published in: Duke Mathematical Journal
publication date: 1990-01-01T00:00:00Z
review text: Let X denote the flag variety associated to a semi-simple complex group G. For an element w of the Weyl group of G let \(L_ w\) denote the irreducible D-module with regular singularities on X supported on the Schubert variety \(\bar X_ w\). The following conjecture is due to \textit{Kazhdan-Lusztig}: ``For \(G=SL_ n\), the singular support of \(L_ w\) is an irreducible subvariety of \(T^*X''.\) The authors treat the case of Grassmannians (degenerate flag varieties for \(SL_ n)\) and prove the following theorem: ``Let \(G_{k,\ell}\) denote the Grassmannian of k-planes in \(k+\ell\) dimensional complex vector space. Let \(\bar X_{\lambda}\) be a Schubert variety in \(G_{k,\ell}\). Let \(L_{\lambda}\) denote the irreducible holonomic D- module with regular singularities on \(G_{k,\ell}\) supported on \(\bar X_{\lambda}\). Then the singular support of \(L_{\lambda}\) is an irreducible subvariety of \(T^*G_{k,\ell}''.\) The theorem is reduced to study of Zelevinsky's resolutions using the formula of Brylinski for calculation of multiplicities of components of the characteristic cycle of a D-module in terms of vanishing cycles. Then, the proof is based on a detailed analysis of the geometry of the resolutions of singularities of Schubert varieties in Grassmannians introduced by Zelevinsky.
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 14C25
Mathematics Subject Classification ID: 14E15
Mathematics Subject Classification ID: 32S40
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 4200385
zbMATH Keywords: flag variety
zbMATH Keywords: Grassmannians
zbMATH Keywords: multiplicities of components of the characteristic cycle of a D-module
zbMATH Keywords: vanishing cycles
zbMATH Keywords: resolutions of singularities
zbMATH Keywords: Schubert varieties
author: Paul Bressler
author: Michael Finkelberg
author: Valery A. Lunts
reviewed by: Vasile Brînzănescu
MaRDI profile type: Publication
cites work: Q3762606
cites work: Characteristic varieties and vanishing cycles
cites work: Intersection homology. II
cites work: A topological approach to Springer's representations
cites work: Q3481902
cites work: Small resolutions of singularities of Schubert varieties
cites work: Q4740076
description: scientific article; zbMATH DE number 4200385