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Stable Bi-Period Summation Formula and Transfer Factors

  Published:1999-08-01
 Printed: Aug 1999
  • Yuval Z. Flicker
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Abstract

This paper starts by introducing a bi-periodic summation formula for automorphic forms on a group $G(E)$, with periods by a subgroup $G(F)$, where $E/F$ is a quadratic extension of number fields. The split case, where $E = F \oplus F$, is that of the standard trace formula. Then it introduces a notion of stable bi-conjugacy, and stabilizes the geometric side of the bi-period summation formula. Thus weighted sums in the stable bi-conjugacy class are expressed in terms of stable bi-orbital integrals. These stable integrals are on the same endoscopic groups $H$ which occur in the case of standard conjugacy. The spectral side of the bi-period summation formula involves periods, namely integrals over the group of $F$-adele points of $G$, of cusp forms on the group of $E$-adele points on the group $G$. Our stabilization suggests that such cusp forms---with non vanishing periods---and the resulting bi-period distributions associated to ``periodic'' automorphic forms, are related to analogous bi-period distributions associated to ``periodic'' automorphic forms on the endoscopic symmetric spaces $H(E)/H(F)$. This offers a sharpening of the theory of liftings, where periods play a key role. The stabilization depends on the ``fundamental lemma'', which conjectures that the unit elements of the Hecke algebras on $G$ and $H$ have matching orbital integrals. Even in stating this conjecture, one needs to introduce a ``transfer factor''. A generalization of the standard transfer factor to the bi-periodic case is introduced. The generalization depends on a new definition of the factors even in the standard case. Finally, the fundamental lemma is verified for $\SL(2)$.
MSC Classifications: 11F72, 11F70, 14G27, 14L35 show english descriptions Spectral theory; Selberg trace formula
Representation-theoretic methods; automorphic representations over local and global fields
Other nonalgebraically closed ground fields
Classical groups (geometric aspects) [See also 20Gxx, 51N30]
11F72 - Spectral theory; Selberg trace formula
11F70 - Representation-theoretic methods; automorphic representations over local and global fields
14G27 - Other nonalgebraically closed ground fields
14L35 - Classical groups (geometric aspects) [See also 20Gxx, 51N30]
 

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