L-Functions and the Oscillator Representation [electronic resource] / by Stephen Rallis.
Material type: TextSeries: Lecture Notes in Mathematics ; 1245Publisher: Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1987Description: XVI, 240 p. online resourceContent type:- text
- computer
- online resource
- 9783540477617
- 512.7 23
- QA241-247.5
Notation and preliminaries -- Special Eisenstein series on orthogonal groups -- Siegel formula revisited -- Inner product formulae -- Siegel formula — Compact case -- Local l-factors -- Global theory.
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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