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
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1245 Stephen Rallis
L-Functions and the Oscillator Representation
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Author
Stephen Rallis Department of Mathematics The Ohio State University Columbus, Ohio 43210, USA
Mathematics Subject Classification (1980): 10DXX
ISBN 3-540-17694-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-17694-2 Springer-Verlag New York Berlin Heidelberg
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Introducti on The purpose of this work is to show how the recent work of Waldspurger ([W-l] and [W-2]) can be fit into a general theory of usi ng the osci 11 ator representat ion to
construct certai n l- funct ions of automorphic representati ons
(l)
associated to classical groups and (2)
give exact formulae for the special
l-functions at integer and half integer points.
values of these
We are
presenting a point of view that is the outgrowth of our results in [R-2] and [R-3].
We note that this work was initiated in an attempt to relate the results
of [R-3] to Waldspurger's results in [W-2]. We describe the general idea. Indeed we start with a dual reductive pair, oscillator representation of on G' (A)
and a general
G x G on a Schwartz space
6 -kernel .p
(.p
and the associated S.
With a cusp form f
a Schwartz function), we can construct
(,
,f
G xG
> we get a formula of
')()' fA
The main problem here is to interpret the second term above. tensor product of
(5 0 5)
(G x G,G' x G').
dual pair
=S
and consider the associated oscillator representation
However it is possible to apply Kudla's idea of a see-saw so that G' Moreover we have the
[Ku-2]; that is, there exists another dual pair
A = {(g,g)lg E G} and so that G fundamental relationship (3)
First we take the
acts on
GjG>}
;, f
is an inner
IX
product associated to a lifting of
f
to a smaller dimensional
O(Q').
Thus
assuming knowl edge of the lower rank cases it suf f i ces to exami ne the fi rst term of the sum above.
The analytic family
of Eisenstein series is constructed s this parabolic from data relative to a maximal parabolic subgroup of Sp n subgroup has Levi factor of the form Sp is formed from the x GR.. , and JE s n-i tensor product of a Siegel-Weil integral (relative to a dual pai r of form JE
,
O(Q')
Sp
x
.)
n-l
and a one dimensional character on GR.,..
It is precisely at this
point that we apply our v
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