Harmonic polynomials and Springer's representation for \(\text{SL}(n,\mathbb{C})\)
type: Item
instance of: scholarly article
title: Harmonic polynomials and Springer's representation for \(\text{SL}(n,\mathbb{C})\)
zbMATH Open document ID: 0827.20053
DOI: 10.1016/0021-8693(92)90164-H
author: Michela Varagnolo
published in: Journal of Algebra
publication date: 1993-03-10T00:00:00Z
review text: Let \(G\) be a connected reductive complex algebraic group and let \(\mathfrak g\) be its Lie algebra. Let \(B\) be the variety of all Borel subgroups of \(G\). This paper only deals with the case \({\mathfrak g} = \text{sl}(n,\mathbb{C})\). In the special case \({\mathfrak g} = \text{sl}(n,\mathbb{C})\), \(B\) becomes isomorphic to the variety \(T = T(V)\) of all complete flags in an \(n\)-dimensional complex space \(V\) and the nilpotent orbits are in bijective correspondence with partitions of \(n\). Let \(u\) be a unipotent linear transformation of \(V\) of Jordan partition \(\sigma = (\sigma_1 \geq \sigma_2 \geq \dots \geq \sigma_m)\). Then the Springer variety, denoted by \(T_\sigma\), is the fixed point subvariety of \(T\) with respect to the action of \(u\). Let \(\Delta\) be the root system of the Weyl group \(S_\mu = S_{\mu_1} \times \dots \times S_{\mu_r}\) and \(\mu = (\mu_1 \geq \dots \geq \mu_r)\). Clearly, for the Weyl groups \(S_\mu\), the Weyl dimension polynomial \(p_\Delta\) is \(S_\mu\)-harmonic. Using results of \textit{S. G. Hulsurkar} [Invent. Math. 27, 45-52 (1974; Zbl 0298.17005)], an explicit expression of the homology \(H_* (T_\sigma, \mathbb{Q})\) is given in terms of the Weyl dimension polynomials \(p_\Delta\).
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 22E45
Mathematics Subject Classification ID: 20G20
Mathematics Subject Classification ID: 17B20
zbMATH DE Number: 130532
zbMATH Keywords: harmonic polynomials
zbMATH Keywords: connected reductive complex algebraic group
zbMATH Keywords: Lie algebra
zbMATH Keywords: Borel subgroups
zbMATH Keywords: nilpotent orbits
zbMATH Keywords: Springer variety
zbMATH Keywords: homology
reviewed by: Zhonghu Chen
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 130532