Selberg zeta-functions for cofinite lattices acting on line bundles over complex hyperbolic spaces
type: Item
instance of: scholarly article
title: Selberg zeta-functions for cofinite lattices acting on line bundles over complex hyperbolic spaces
zbMATH Open document ID: 1096.11035
author: Khadija Ayaz
published in: Hiroshima Mathematical Journal
publication date: 2005-05-06T00:00:00Z
review text: For a homogeneous bundle \(E\) on a locally symmetric space of rank one \(Y=\Gamma\backslash G/K\) (\(G\) semi-simple Lie group of \(\mathbb R\)-rank one, \(K\) compact maximal subgroup of \(G\) and \(\Gamma\) discrete subgroup of \(G\) without elliptic elements), the Selberg zeta-function \(Z_{\Gamma, E}\) encodes the holonomy spectrum of its periodic geodesics. The case with \(Y\) compact is rather well understood, either by the dynamical approach of \textit{D. Fried} [Ann. Sci. Éc. Norm. Supér. (4) 19, 491--517 (1986; Zbl 0609.58033)] or by harmonic analysis through a trace formula [e. g. \textit{U. Bunke} and \textit{M. Olbrich}, Selberg zeta and theta functions. A differential operator approach. Mathematical Research 83 (Akademie Verlag, Berlin) (1995; Zbl 0831.58001)]. For \(Y\) non-compact but of finite volume, the dynamical approach failed because of the non-uniform dynamical hyperbolicity in the cusps: nevertheless a trace formula does exist as proved by \textit{G. Warner} [Adv. Math. Suppl. Stud. 6, 1--142 (1979; Zbl 0466.10018)]. Its terms induced by the continuous spectrum require a special analysis, which has been done only in a few cases, e.g. for \(4d\)-dimensional complex hyperbolic manifolds by \textit{R. Gangolli} and \textit{G. Warner} [J. Math. Soc. Japan 27, 328--343 (1975; Zbl 0325.22014)]. The author considers here the case of some homogeneous line bundle \(E_\ell\) on any complex hyperbolic space : \(G=SU(n,1)\) and the bundle \(E_\ell\) is given by the \(1\)-dimensional \(\tau_\ell\) representation defined by \(\tau_\ell(\sigma,z)=z^\ell, (\sigma,z)\in K=S(U(n)\times U(1))\). Using the harmonic analysis tools established by \textit{P. C. Trombi} for \(\tau_\ell\)-spherical functions [Pac. J. Math. 101, 223--245 (1982; Zbl 0572.22005)], the author writes down the trace formula for an appropriate test function depending on a parameter \(s\). A Selberg zeta-function \(Z_{\gamma,\ell}\) is defined through its logarithmic derivative on the half-p…
reviewed by: Laurent Guillopé
Mathematics Subject Classification ID: 11M36
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 22E45
zbMATH DE Number: 2165948
zbMATH Keywords: Selberg zeta-function
zbMATH Keywords: spectral theory
zbMATH Keywords: trace formula
zbMATH Keywords: spherical functions
zbMATH Keywords: complex hyperbolic space
zbMATH Keywords: Abel transform
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.32917/hmj/1150998509
OpenAlex ID: W1489400755
Wikidata QID: Q128872355
DOI: 10.32917/HMJ/1150998509
Recommended article: The Selberg trace formula and the Selberg zeta-function for cocompact discrete subgroups of \(SO_ +(1,n)\)
Recommended article: The twisted Selberg trace formula and the twisted Selberg zeta function for compact orbifolds
Recommended article: Q4344092
Recommended article: Selberg and Ruelle zeta functions for non-unitary twists
Recommended article: Selberg zeta functions on odd-dimensional hyperbolic manifolds of finite volume
Recommended article: A note on the Selberg zeta function for compact quotients of hyperbolic spaces
Recommended article: Q4848613
Recommended article: The Selberg trace formula and the Ruelle zeta function for compact hyperbolics
Recommended article: The divisor of Selberg's zeta function for Kleinian groups. Appendix A by Charles Epstein
Recommended article: Q3739218
description: scientific article; zbMATH DE number 2165948