Springer's Weyl group representations through characteristic classes of cone bundles
type: Item
instance of: scholarly article
title: Springer's Weyl group representations through characteristic classes of cone bundles
zbMATH Open document ID: 0624.14015
DOI: 10.1007/BF01458071
author: Robert MacPherson
author: Walter Borho
author: Jean-Luc Brylinski
published in: Mathematische Annalen
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164292
review text: A nilpotent orbit \({\mathcal O}_ u\) in a complex semisimple Lie algebra gives rise to a collection of cone bundles on the flag variety, by taking the closed components of its preimage under Springer's resolution of singularities. Using the generalization of inverse Chern classes of vector bundles to Segre classes of cone bundles due to Fulton and the third author, we attach to each such cone bundle a characteristic class in the cohomology of the flag variety, which is interpreted as a harmonic polynomial on the Cartan subalgebra. Using the intersection homology approach to the study of nilpotent varieties as by \textit{W. Borho} and \textit{R. MacPherson} in Astérisque 101-102, 23-74 (1983; Zbl 0576.14046) and C. R. Acad. Sci., Paris, Sér. I 292, 707-710 (1981; Zbl 0467.20036) we show that this collection of polynomials transforms under the action of the Weyl group according to Springer's irreducible representation \(\rho_ u\) which is usually constructed from \({\mathcal O}_ u\) by quite different means.
Mathematics Subject Classification ID: 14F99
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 14F05
zbMATH DE Number: 4012802
zbMATH Keywords: Weyl group representations
zbMATH Keywords: cone bundles on the flag variety
zbMATH Keywords: characteristic class
zbMATH Keywords: cohomology of the flag variety
zbMATH Keywords: intersection homology
zbMATH Keywords: action of the Weyl group
Wikidata QID: Q101423854
MaRDI profile type: Publication
cites work: On Springer's correspondence for simple groups of type En (n = 6, 7, 8)
cites work: Riemann-Roch and topological K-theory for singular varieties
cites work: Q3320507
cites work: Green functions of finite Chevalley groups of type \(E_ n (n=6,7,8)\)
cites work: Topics in the homology theory of fibre bundles. Lectures given at the University of Chicago, 1954. Notes by E. Halpern
cites work: Differential operators on homogeneous spaces. I: Irreducibility of the associated variety for annihilators of induced modules
cites work: Differential operators on homogeneous spaces. III: Characteristic varieties of Harish Chandra modules and of primitive ideals
cites work: Q3697198
cites work: Q3919888
cites work: Q3697154
cites work: Q3328651
cites work: Q4165449
cites work: Intersection homology theory
cites work: Intersection homology. II
cites work: Q3942064
cites work: Q3708976
cites work: On Joseph's construction of Weyl group representations
cites work: On the variety of a highest weight module
cites work: A topological approach to Springer's representations
cites work: Green polynomials and singularities of unipotent classes
cites work: LECTURES ON THEK-FUNCTOR IN ALGEBRAIC GEOMETRY
cites work: On the springer representations of the weyl groups of classical algebraic groups
cites work: On the springer representations of chevalley groups op type f4
cites work: Q3860031
cites work: On the fixed point set of a unipotent element on the variety of Borel subgroups
cites work: Trigonometric sums, Green functions of finite groups and representations of Weyl groups
cites work: A construction of representations of Weyl groups
cites work: On the desingularization of the unipotent variety
OpenAlex ID: W1964916660
Recommended article: Q3760759
Recommended article: Springer’s Weyl Group Representation via Localization
Recommended article: Weyl group actions on the Springer sheaf
Recommended article: From families in Weyl groups to Springer representations
Recommended article: Spin representations of Weyl groups and the Springer correspondence
Recommended article: Springer's work on unipotent classes and Weyl group representations
Recommended article: On Springer's representations of Weyl groups containing -1
Recommended article: Q3731761
Recommended article: Springer representations on the Khovanov Springer varieties
Recommended article: Characters of Springer representations on elliptic conjugacy classes
description: scientific article; zbMATH DE number 4012802