Line bundles on the cotangent bundle of the flag variety
type: Item
instance of: scholarly article
title: Line bundles on the cotangent bundle of the flag variety
zbMATH Open document ID: 0807.14043
DOI: 10.1007/BF01244299
published in: Inventiones Mathematicae
publication date: 1995-03-05T00:00:00Z
full work available at URL: https://eudml.org/doc/144119
review text: Let \(G\) be a reductive group defined over an algebraically closed field \(k\). Choose a Borel subgroup \(B\) containing a maximal torus \(T\). Every character \(\chi\) of \(T\) extends to a character of \(B\) and gives rise to an invertible sheaf \({\mathcal L}_{G/B} (k_ \chi)\). The main result of the present article is: The pull back \({\mathcal L}_{\mathcal T} (k_ \chi)\) of \({\mathcal L}_{G/B} (k_ \chi)\) to the cotangent bundle \({\mathcal T}\) of \(G/B\) has vanishing cohomology if and only if there is no dominant weight strictly between \(\chi\) and \(\chi^ +\) in the Chevalley order, where \(\chi^ +\) is the dominant weight in the Weyl group orbit of \(\chi\). This result corrects a vanishing statement of \textit{H. H. Andersen} and \textit{J. C. Jantzen} [Math. Ann. 269, 487-525 (1984) = Prepr. Ser., Aarhus Univ. 1983/84, No. 34 (1984; Zbl 0529.20027)], and together with the following result, generalizes results of \textit{W. H. Hesselink} [Math. Ann. 223, 249-252 (1976; Zbl 0318.14007) and 252, 179-182 (1980; Zbl 0447.17006)]. In the case of vanishing, the global sections of \({\mathcal L}_{\mathcal T} (k_ \chi)^*\) and the global sections \({\mathcal J} (\chi^ +)\) of \({\mathcal L}_{\mathcal T} (k_{\chi^ +})^*\) are isomorphic after a shift of degrees. Another main result of the article is: \({\mathcal J} (\chi^ +)\) considered as an \(k[{\mathfrak g}]\) module is generated by its elements of degree zero, forming a simple \(G\)-module of type \(V_{\chi^ +}^*\). The latter result implies a conjecture of \textit{R. K. Brylinski}. The above results have many important and interesting applications, some of which had already been pointed out by Brylinski. An application is to the study of the structure of the subregular nilpotent variety \({\mathcal S}\). When \(G\) is simple and \(\varphi\) its short dominant root, the author proves that \({\mathcal J} (\varphi)\), after a shift in degrees, is the prime ideal defining \({\mathcal S}\) in the coordinate ri…
reviewed by: Dan Laksov
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 14F05
zbMATH DE Number: 549678
zbMATH Keywords: flag variety
zbMATH Keywords: Borel subgroup
zbMATH Keywords: cotangent bundle
zbMATH Keywords: vanishing cohomology
zbMATH Keywords: Gorenstein variety
zbMATH Keywords: rational singularities
author: Abraham Broer
MaRDI profile type: Publication
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OpenAlex ID: W1966027335
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description: scientific article; zbMATH DE number 549678