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Search: MSC category 14L35 ( Classical groups (geometric aspects) [See also 20Gxx, 51N30] )

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1. CJM 1999 (vol 51 pp. 771)

Flicker, Yuval Z.
 Stable Bi-Period Summation Formula and Transfer Factors 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)\$. Categories:11F72, 11F70, 14G27, 14L35

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