Perturbation theory for twisted automorphic functions
type: Item
instance of: scholarly article
title: Perturbation theory for twisted automorphic functions
zbMATH Open document ID: 0871.11039
DOI: 10.1007/PL00001614
published in: Geometric and Functional Analysis. GAFA
publication date: 1997-04-16T00:00:00Z
review text: Let \(\Gamma\) be a cofinite non-cocompact discrete subgroup of \(\text{SL}(2,\mathbb{R})\) and \(\rho\) a finite-dimensional irreducible unitary representation of \(\Gamma\) acting on a \(d\)-dimensional unitary vector space. \((\Gamma,\rho)\) is called essentially cuspidal if the cusp form eigenvalues for \((\Gamma,\rho)\) embedded in the continuous spectrum satisfy Weyl's law. The purpose of the paper under review is to show that the generic \((\Gamma,\rho)\) is not essentially cuspidal if one assumes for the trivial representation that the dimension of the eigenspaces containing at least one newform is bounded. The present paper extends previous work of \textit{R. Phillips} and \textit{P. Sarnak} [Geom. Funct. Anal. 4, 93-118 (1994; Zbl 0804.11038)] to cover the case of multidimensional irreducible unitary representations under the premise quoted above. To simplify the exposition the author limits himself to the case \(\Gamma=\Gamma(2)\), the principal congruence subgroup of level 2 in \(\text{SL}_2(\mathbb{Z})\). The author's approach to the problem is based on perturbation theory.
reviewed by: Jürgen Elstrodt
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 1000117
zbMATH Keywords: twisted automorphic functions
zbMATH Keywords: Laplacian
zbMATH Keywords: essentially cuspidal
zbMATH Keywords: multidimensional irreducible unitary representations
zbMATH Keywords: perturbation theory
author: Ralph S. Phillips
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1000117