Lefschetz numbers and twisted stabilized orbital integrals
type: Item
instance of: scholarly article
title: Lefschetz numbers and twisted stabilized orbital integrals
zbMATH Open document ID: 0808.11039
DOI: 10.1007/BF01445102
published in: Mathematische Annalen
publication date: 1995-03-09T00:00:00Z
full work available at URL: https://eudml.org/doc/165082
review text: Consider a connected simply connected semisimple algebraic group \(G\) defined over \(\mathbb{Q}\) such that \(G(R)\) is non-compact. Let \(\sigma\) be a \(\mathbb{Q}\)-rational automorphism of \(G\) of finite order, and \(\Gamma\) an arithmetic, \(\sigma\)-stable and torsion-free subgroup of \(G(\mathbb{Q})\). For a complex finite-dimensional representation on a space \(V\) with compatible \(\sigma\)-action we can define an action \(\sigma^ i\) of \(\sigma\) on the cohomology group \(H^ i (\Gamma,V)\) and then define a Lefschetz number \(L(\sigma, \Gamma, V)=\sum (-1)^ i \text{tr } \sigma^ i\), where the sum is taken from \(i=0\) to infinity. The main result of the paper is an explicit formula for the Lefschetz number. As claimed by the authors this formula has the following applications: Computationally, it provides a method to compute the actual Lefschetz number. Topologically, it can be viewed as a \(\sigma\)-twisted variant of the Hirzebruch proportionality principle; the untwisted version of this principle is in this case Harder's Gauss-Bonnet formula. Analytically, it fits into the picture of the conjectural twisted stabilization of the Arthur-Selberg trace formula.
Mathematics Subject Classification ID: 11F75
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 537294
zbMATH Keywords: orbital integral
zbMATH Keywords: semisimple algebraic group
zbMATH Keywords: cohomology group
zbMATH Keywords: explicit formula
zbMATH Keywords: Lefschetz number
zbMATH Keywords: Hirzebruch proportionality principle
zbMATH Keywords: Harder's Gauss-Bonnet formula
zbMATH Keywords: Arthur-Selberg trace formula
author: Jürgen Rohlfs
author: Birgit Speh
reviewed by: Yangbo Ye
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 537294