Spectral deformations of automorphic functions over graphs of groups
type: Item
instance of: scholarly article
title: Spectral deformations of automorphic functions over graphs of groups
zbMATH Open document ID: 0707.11041
DOI: 10.1007/BF01233435
author: Isaac Efrat
published in: Inventiones Mathematicae
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143842
review text: Let X be a space on which a group \(\Gamma\) acts isometrically, and let \(\Gamma '\) be a normal subgroup of finite index. Then under wide assumptions the space \(L^ 2(\Gamma '\setminus X)\) splits into the direct sum of the spaces \(L^ 2(\Gamma \setminus X,\rho)^{\deg \rho}\) where \(\rho\) ranges over the irreducible representations of \(\Gamma '\setminus \Gamma\). A fundamental question in the spectral theory of automorphic forms is then, whether \(L^ 2(\Gamma '\setminus X)\) is just what stems from \(L^ 2(\Gamma \setminus X)\) or whether the spaces \(L^ 2(\Gamma \setminus X,\rho)\) make an additional contribution and what this contribution may be. This problem was discussed by \textit{P. Sarnak} (cf. the preceding review), in the case of automorphic forms on hyperbolic spaces. In particular, Sarnak investigated the disappearance of discrete spectra under deformations and this made him conjecture that there should exist groups whose spectrum is abundant in \(L^ 2(\Gamma \setminus X)\) and is nevertheless destroyed by a deformation. The aim of the paper under review is to prove this conjecture in the context of the Bruhat-Tits building X attached to \(GL_ 2(K)\) where K is the completion at infinity of the function field of genus 0 over \({\mathbb{F}}_ q\). Let \(\Gamma\) be a discrete subgroup of the group of isometries of X such that \(\Gamma\setminus X\) is of finite volume. The square-integrable eigenfunctions of the invariant operators on X span a finite-dimensional space in \(L^ 2(\Gamma \setminus X)\). Neglecting the trivial two-dimensional part of this space generated by the constant and alternating functions let \(L^ 2(\Gamma \setminus X)_{dis}\) be the non-trivial part. Main Theorem: For every integer \(N>0\) there exists a group \(\Gamma =\Gamma_ N\), such that dim \(L^ 2(\Gamma \setminus X)_{dis}=N\), yet dim \(L^ 2(\Gamma \setminus X,\chi)_{dis}=0\) for every character \(\chi\neq 1\) in a neighbourhood of the identity character. Hence, in strong cont…
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20E42
Mathematics Subject Classification ID: 20B27
Mathematics Subject Classification ID: 20E08
zbMATH DE Number: 4160850
zbMATH Keywords: spectral deformation
zbMATH Keywords: graph of groups
zbMATH Keywords: automorphic forms
zbMATH Keywords: Bruhat-Tits building
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4160850