L-functions and the oscillator representation / Stephen Rallis

The theory of dual reductive pairs and the oscillator (Weil) representation yields ’correspondences’ between the representations of these pairs over both local and global fields. The local theory is now fairly well understood but the global theory turns out to be more subtle. Around 1980 J.-L. Waldspurger demonstrated that the image of an automorphic representation under the Shimura correspondence is zero or not depends on, apart from local conditions, whether a certain L-function vanishes at a special point or not.

In this book the author proposes a much broader framework in which to understand Waldspurger’s theorem, and which leads to a very wide generalization. He also proposes novel methods of proof. Here this programme is described and carried through for a restricted class of dual reductive pairs; this class contains the case considered by Waldspurper but is much wider. The technique used here involves computing the L2 -norm of elements of the image under the correspondence and relating this to the Siegel theorem for a dual reductive pair related to the original one. (Zentralblatt)