L-Functions and the Oscillator Representation

1. Front Matter

   Pages I-XIV

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2. Notation and preliminaries

   • Stephen Rallis

   Pages 1-9

3. Special Eisenstein series on orthogonal groups

   • Stephen Rallis

   Pages 10-24

4. Siegel formula revisited

   • Stephen Rallis

   Pages 25-48

5. Inner product formulae

   • Stephen Rallis

   Pages 49-86

6. Siegel formula — Compact case

   • Stephen Rallis

   Pages 87-127

7. Local l-factors

   • Stephen Rallis

   Pages 128-173

8. Global theory

   • Stephen Rallis

   Pages 174-199

9. Back Matter

   Pages 200-239

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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

• automorphic forms
• lie group
• representation theory

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