On the weighted fundamental lemma for twisted endoscopy of the general linear group

Abstract: We present a conjecture asserting an identity between stable weighted orbital integrals on the groups Sp(2n) and SO(2n+1). We show that this conjecture, together with the weighted fundamental lemma for standard endoscopy, implies the twisted weighed fundamental for GL(n) with respect to the transpose inverse automorphism. In particular this allows us to deduce, from results in the literature, the twisted weighted fundamental lemma necessary to understand the lifting of automorphic representations from SO(5) to GL(4) and from Sp(4) to GL(5).

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