TY - JOUR

T1 - Stable bi-period summation formula and transfer factors

AU - Flicker, Yuval Z.

PY - 1999/8

Y1 - 1999/8

N2 - 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 ⊗ 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 endoscopie 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 endoscopie 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).

AB - 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 ⊗ 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 endoscopie 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 endoscopie 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).

UR - http://www.scopus.com/inward/record.url?scp=0033413449&partnerID=8YFLogxK

U2 - 10.4153/CJM-1999-033-9

DO - 10.4153/CJM-1999-033-9

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AN - SCOPUS:0033413449

SN - 0008-414X

VL - 51

SP - 771

EP - 791

JO - Canadian Journal of Mathematics

JF - Canadian Journal of Mathematics

IS - 4

ER -