This book studies the modules arising in Fourier expansions of automorphic forms, namely Fourier term modules on SU(2,1), the smallest rank one Lie group with a non-abelian unipotent subgroup. It considers the abelian Fourier term modules connected to characters of the maximal unipotent subgroups of SU(2,1), and also the non-abelian modules, described via theta functions. A complete description of the submodule structure of all Fourier term modules is given, with a discussion of the consequences for Fourier expansions of automorphic forms, automorphic forms with exponential growth included. These results can be applied to prove a completeness result for Poincaré series in spaces of square integrable automorphic forms. Aimed at researchers and graduate students interested in automorphic forms, harmonic analysis on Lie groups, and number-theoretic topics related to Poincaré series, the book will also serve as a basic reference on spectral expansion with Fourier-Jacobi coefficients. Only a background in Lie groups and their representations is assumed.
Introduction.- The Lie group SU(2,1) and subgroups.- Fourier term modules.- Submodule structure.- Application to automorphic forms.
Roelof W. Bruggeman was born in Zwolle, the Netherlands. He obtained his PhD at Utrecht University in 1972, and was a postdoctoral fellow at Yale University (197273). He has worked at Utrecht University since 1980, now as a guest after his retirement in 2005. In 2022 he became a corresponding member of the Academia Nacional de Ciencias in Córdoba, Argentina. The main research themes in his work are the spectral theory of Maass forms, the study of families of automorphic forms as a function of complex parameters, and the relation between automorphic forms and cohomology. Roberto J. Miatello was born in Buenos Aires, Argentina. After studying at FaMAF, UNCordoba, Argentina (19651970) he obtained his PhD from Rutgers University in 1976. He was a professor at UFPe, Brazil (1977-1980), a member of the IAS, Princeton, USA (1980-81), and has been a professor at FaMAF (UNC) since 1982, becoming Professor Emeritus in 2016. He is a member of Conicet and (since 1996) the National Academy of Sciences of Argentina. His research focuses on geometry, the spectral theory of locally symmetric varieties, and automorphic forms.
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