Continuous spectrum for SL 2 ( Z )

Paul Garrett [email protected] http://www.math.umn.edu/ g̃arrett/ [This document is http://www.math.umn.edu/ ̃garrett/m/mfms/notes 2013-14/13 1 cont afc spec.pdf] 1. Pseudo-Eisenstein series adjunction to constant term 2. Decomposition of pseudo-Eisenstein series: beginning 3. Eisenstein series 4. Decomposition of pseudo-Eisenstein series: conclusion 5. Plancherel theorem We prove that the orthogonal complement in L(Γ\H) to cuspforms is spanned by pseudo-Eisenstein series, in turn expressible as integrals or wave packets of Eisenstein series Es. We prove a Plancherel theorem for this space. The essential harmonic analysis is Fourier transform on the real line, in coordinates on which it is called a Mellin transform. That is, the non-cuspidal part of harmonic analysis on SL2(Z)\H reduces to harmonic analysis on a related, lower-dimensional group.