L Functions and the Oscillator Representation 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