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The fundamental Lemma of Jacquet and Rallis. Appendix by Julia Gordon. (English) Zbl 1211.14039
H. Jacquet and S. Rallis in their manuscript ‘On the Gross-Prosad conjecture for the unitary group in 3 variables’ [http://www.math.columbia.edu/~hj] proposed an approach to the Gross-Prosad conjecture for the unitary groups using the relative trace formula. In establishing the relative trace formula, they needed a form of the fundamental lemma comparing the orbital integrals of the standard test functions on the symmetric space \(GL_n(E)/GL_n(F)\) and on the unitary group \(U_n(F)\), where \(E/F\) is an unramified extension of a local field \(F\) with odd residue characteristic. They explicitly stated (up to a sign) a Lie algebra version of this fundamental lemma as a conjecture and verified it for \(n\geq 3\).
Following this idea, W. Zhang in the manuscript ‘On the ramified Gross-Prosad conjecture for unitary groups in three variables’ [preprint (2009)] stated the group version of this fundamental lemma as a conjecture and verified it for \(n\geq 3\).
The purpose of the present article is to prove the above conjectures in the case when \(F\) is a local function filed and char\((F)>n\). The proof follows the strategy of the proof of Langlands-Shelstad fundamental lemma in the Lie algebra and function field case.
In the appendix written by Julia Gordon, it is shown that the transfer principle of Cluckers and Loeser, which relies on model-theoretical methods, applies to the present situation too. Therefore, the results proved for local function fields imply the validity of the above conjectures for any local field of sufficiently large characteristic.

MSC:
14H60 Vector bundles on curves and their moduli
22E35 Analysis on \(p\)-adic Lie groups
14F20 Étale and other Grothendieck topologies and (co)homologies
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