Zero weight spaces and the Springer correspondence
type: Item
instance of: scholarly article
title: Zero weight spaces and the Springer correspondence
zbMATH Open document ID: 0921.22012
DOI: 10.1016/S0019-3577(98)80010-4
author: Mark Reeder
published in: Indagationes Mathematicae. New Series
publication date: 1999-09-26T00:00:00Z
review text: The irreducible representations of \({\text{SL}}(n,{\mathbb C})\) are parameterised by Young tableau. If the tableau has \(n\) boxes, then the representation has the property that zero is a weight but twice a root is not a weight. For any semisimple group, a representation with this property is called `small'. It may be checked for \({\text{SL}}(n,{\mathbb C})\), that the action of the symmetric group \(S_n\) (as the Weyl group of \({\text{SL}}(n,{\mathbb C})\)) on the zero weight space of a small representation is the representation of \(S_n\) given by the dual tableau. This article generalises this fact to the other simply laced groups, the \(D\) and \(E\) series. For this, we need to give another description of the action of \(S_n\) corresponding to a particular tableau. It is the Springer representation determined as follows. Choose a unipotent element \(u\) in \({\text{SL}}(n,{\mathbb C})\) with Jordan blocks of sizes determined by the tableau regarded as a partition of \(n\). Then the representation is realised as the top cohomology with complex coefficients of the fixed point set of \(u\) in the full flag variety of \({\text{SL}}(n,{\mathbb C})\). It is in this form that the statement for \({\text{SL}}(n,{\mathbb C})\) is generalised. It asserts the existence of a unipotent element \(u\in G\) so that exactly the same statement holds. There is a difference in that these Weyl group representations may be reducible. The proof is empirical, i.e. by determining all small representations and so on. For \(G\) that are not simply laced, this particular formulation fails and it is an interesting question whether there is a generally valid reformulation.
reviewed by: M. Eastwood
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 20C30
zbMATH DE Number: 1251203
zbMATH Keywords: Lie group
zbMATH Keywords: representation
zbMATH Keywords: Springer representation
zbMATH Keywords: weight space
zbMATH Keywords: Young tableau
zbMATH Keywords: semisimple group
zbMATH Keywords: symmetric group
zbMATH Keywords: flag variety
MaRDI profile type: Publication
cites work: Q3803334
cites work: Some numerical results on the characters of exceptional Weyl groups
cites work: The sum of generalized exponents and Chevalley's restriction theorem for modules of covariants
cites work: Q3222341
cites work: Q3682663
cites work: Lie Group Representations on Polynomial Rings
cites work: Exterior Powers of the Adjoint Representation
cites work: Invariants of Finite Reflection Groups
description: scientific article; zbMATH DE number 1251203