L-Functions and the Oscillator Representation

by Stephen Rallis

Overview

These notes are concerned with showing the relation between L-functions of classical groups (*F1 in particular) and *F2 functions arising from the oscillator representation of the dual reductive pair *F1 *F3 O(Q). The problem of measuring the nonvanishing of a *F2 correspondence by computing the Petersson inner product of a *F2 lift from *F1 to O(Q) is considered. This product can be expressed as the special value of an L-function (associated to the standard representation of the L-group of *F1) times a finite number of local Euler factors (measuring whether a given local representation occurs in a given oscillator representation). The key ideas used in proving this are (i) new Rankin integral representations of standard L-functions, (ii) see-saw dual reductive pairs and (iii) Siegel-Weil formula. The book addresses readers who specialize in the theory of automorphic forms and L-functions and the representation theory of Lie groups. N

Product Details

ISBN-13:	9783540176947
Publisher:	Springer Berlin Heidelberg
Publication date:	05/05/1987
Series:	Lecture Notes in Mathematics , #1245
Edition description:	1987
Pages:	240
Product dimensions:	6.10(w) x 9.25(h) x 0.02(d)

Table of Contents

Notation and preliminaries.- Special Eisenstein series on orthogonal groups.- Siegel formula revisited.- Inner product formulae.- Siegel formula — Compact case.- Local l-factors.- Global theory.