Automorphic Forms on SL2 (R) [Paperback]

An introduction to the analytic theory of automorphic forms in the case of fuchsian groups.The theory of automorphic forms, which goes back to the work of Poincare and Klein, has been considerably developed and generalized in the last 40 years by the use of new analytic methods, inspired in part by harmonic analysis on Lie groups. This book is devoted to the analytic theory of automorphic forms, limited to the case of fuchsian groups, but from those more general points of view.This first exposition of this approach in this special case should be accessible to researchers and graduate students having some familiarity with functional analysis and elementary Lie theory.The theory of automorphic forms, which goes back to the work of Poincare and Klein, has been considerably developed and generalized in the last 40 years by the use of new analytic methods, inspired in part by harmonic analysis on Lie groups. This book is devoted to the analytic theory of automorphic forms, limited to the case of fuchsian groups, but from those more general points of view.This first exposition of this approach in this special case should be accessible to researchers and graduate students having some familiarity with functional analysis and elementary Lie theory.This book provides an introduction to some aspects of the analytic theory of automorphic forms on G=SL2(R) or the upper-half plane X, with respect to a discrete subgroup ^D*G of G of finite covolume. The point of view is inspired by the theory of infinite dimensional unitary representations of G; this is introduced in the last sections, making this connection explicit. The topics treated include the construction of fundamental domains, the notion of automorphic form on ^D*G\G and its relationship with the classical automorphic forms on X, Poincaré series, constant terms, cusp forms, finite dimensionality of the space of automorphic forms of a given type, compactness of certain convolution operators, Eisenstein series, unil³D