Non-minimal modular symbols for GL(n)
type: Item
instance of: scholarly article
title: Non-minimal modular symbols for GL(n)
zbMATH Open document ID: 0671.22005
DOI: 10.1007/BF01388782
author: Avner Ash
published in: Inventiones Mathematicae
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143550
review text: Let M be a Levi factor of a \({\mathbb{Q}}\)-parabolic subgroup P of Gl(n). Let X be the symmetric space for Gl(n) with base point e, \(\Gamma\) an arithmetic subgroup of Gl(n,\({\mathbb{Q}})\), and g any element of Gl(n,\({\mathbb{Q}})\). Then the projection of gM(\({\mathbb{R}})e\) in \(\Gamma\) \(\setminus X\) is a closed submanifold. Its fundamental class defines by Lefschetz duality a cohomology class in \(H^*(\Gamma \setminus X,C)\). This class is called a modular symbol of type M for \(\Gamma\). The main theorem states: Suppose that \(n>1\). Let \(\Gamma\) be a full congruence subgroup of level N, \(N\geq 1\), p a prime that divides N precisely once, and M a Levi factor of a parabolic subgroup with one block of size greater than n/2. Then the span of the modular symbols of M contains no irreducible f-cuspidal representation of SL(n,\({\mathbb{Z}}/p)\).
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 20H05
zbMATH DE Number: 4097680
zbMATH Keywords: Levi factor
zbMATH Keywords: \({\mathbb{Q}}\)-parabolic subgroup
zbMATH Keywords: symmetric space
zbMATH Keywords: arithmetic subgroup
zbMATH Keywords: Lefschetz duality
zbMATH Keywords: cohomology class
zbMATH Keywords: modular symbol
zbMATH Keywords: full congruence subgroup
zbMATH Keywords: block
zbMATH Keywords: irreducible f-cuspidal representation
reviewed by: Birgit Speh
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4097680