L-Functions and the Oscillator Representation
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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...
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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