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Eisenstein series in the domain
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am(y analytic continuation assume automorphic function bounded c(oo closed subspace compact support complete set complex numbers complex variable constant term matrix converges cusp forms cusp oo denote discontinuous group double coset eigenfunction Eisenstein series Eisenstein series E(z element equivalent Er(z Et(z exists finite number finite type reduced follows from Theorem Fourier expansion Fourier transformation functional equation functions with respect fundamental domain Furthermore group of finite Hecke operator Hilbert space holomorphic implies incomplete theta series inequivalent cusps inner product formula invariant integral operator invariant operator irreducible Iwasawa decomposition kernel function Laplacian linear transformation mapping mean value operator Mellin transform modular group non-euclidean obtain orthogonal complement pair invariant k(z Plancherel formula plane H point pair invariant poles proof of Theorem ptj(s q.e.d. Theorem real number reduced at oo resp s)ds s)dx sequence sn set of inequivalent spanned spectral decomposition subgroup theory tion unitary upper half plane Whittaker functions