L-Functions and the Oscillator Representation

Stephen Rallis (auth.)9780387176949, 0-387-17694-2

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

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Table of contents :
Notation and preliminaries….Pages 1-9
Special Eisenstein series on orthogonal groups….Pages 10-24
Siegel formula revisited….Pages 25-48
Inner product formulae….Pages 49-86
Siegel formula — Compact case….Pages 87-127
Local l-factors….Pages 128-173
Global theory….Pages 174-199