Borel-Moore homology and \(K\)-theory on the Steinberg variety
type: Item
instance of: scholarly article
title: Borel-Moore homology and \(K\)-theory on the Steinberg variety
zbMATH Open document ID: 1196.14039
author: Namhee Kwon
published in: Michigan Mathematical Journal
publication date: 2010-02-02T00:00:00Z
review text: For a simply connected complex semisimple Lie group \(G\), let \(\mathfrak g\) be its Lie algebra, \(\mathcal B\) the flag variety of \(G,\) \(\mathcal N\) the nilpotent cone in \(\mathfrak g\), \(\widetilde{\mathcal N} =\{(x,{\mathfrak b}) \in {\mathcal N}\times {\mathcal B}\mid x\in {\mathfrak b}\}\) the Springer resolution of \(\mathcal N,\) and \(Z={\widetilde{\mathcal N}}{\times}_{{\mathcal N}}{\widetilde{\mathcal N}}=\{(x,{\mathfrak b}, {\mathfrak b}') \in {\mathcal N}\times {\mathcal B}\times {\mathcal B}\mid x\in {\mathfrak b}\cap {\mathfrak b}'\}\) the Steinberg variety. In the paper the author studies the \(\mathbb C\)-algebra structure of \(H_*(Z, {\mathbb C})\). Recall that a product on the complex graded Borel Moore homology is given by the convolution product. The study is done by examining multiplications on other objects such as the homology of a flag variety, the Grothendieck group of a graph variety and explicitly constructed by crossed product algebra. This is possible since the author proves the compatibility of the corresponding maps with the products. The main theorem of the paper is the computation of the \(\mathbb C\)-algebra structure of \(H_*(Z, {\mathbb C})\). A similar result was proven independently by \textit{J. M. Douglass} and \textit{G. Röhrle} [J. Algebra 321, No. 11, 3158--3196 (2009; Zbl 1183.20044), Doc. Math., J. DMV 14, 339--357 (2009; Zbl 1209.14020)].
reviewed by: Piotr Krasoń
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 55N35
Mathematics Subject Classification ID: 18F30
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 19L10
Mathematics Subject Classification ID: 57T10
zbMATH DE Number: 5665444
zbMATH Keywords: Steinberg variety
zbMATH Keywords: Borel-Moore homology
zbMATH Keywords: Grothendieck group
zbMATH Keywords: flag variety
zbMATH Keywords: Lie group
zbMATH Keywords: Lie algebra
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1260475700
OpenAlex ID: W2047799734
cites work: Riemann-Roch for singular varieties
cites work: Homology theory for locally compact spaces
cites work: Q2785535
cites work: The Steinberg variety and representations of reductive groups.
cites work: Homology of the Steinberg variety and Weyl group coinvariants
cites work: Categorical framework for the study of singular spaces
cites work: \({\mathfrak G}\)-module, Springer's representations and bivariant Chern classes
cites work: Q3971964
DOI: 10.1307/MMJ/1260475700
Recommended article: Homology of the Steinberg variety and Weyl group coinvariants
Recommended article: On the \(K\)-theory and Hattori-Stallings traces of minimal primitive factors of enveloping algebras of semisimple Lie algebras: the singular case
Recommended article: Hochschild homology of twisted crossed products and twisted graded Hecke algebras
Recommended article: Q3778933
Recommended article: Springer's Weyl group representations through characteristic classes of cone bundles
Recommended article: On Ginzburg’s bivariant Chern classes
Recommended article: Yangians and quantum loop algebras
Recommended article: Q4472990
Recommended article: Q3784990
Recommended article: Q5703394
description: scientific article; zbMATH DE number 5665444