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On geometric transfer in real twisted endoscopy. (English) Zbl 1259.22007
Consider a connected reductive algebraic group \(G\) over \(\mathbb R\) with an automorphism \(\theta\) defined over \(\mathbb R\) and a character \(\omega\) of \(G({\mathbb R})\). The geometric transfer proved in this article is an equality between weighted sums of \((\theta, \omega)\)-twisted orbital integrals on \(G({\mathbb R})\) and stable orbital integrals on an endoscopic group for \((G,\theta,\omega)\). Essential for the proof is the study of the behaviour of stable orbital integrals (resp., of weighted sums of twisted orbital integrals) around a semisimple element with centralizer of type \(A_1\). The author makes use of the fact that the relative transfer factor, i.e., the quotient of two values of the transfer factor, is canonical (see [R. E. Kottwitz and D. Shelstad, Foundations of twisted endoscopy. Astérisque. 255. Paris: Société Mathématique de France (1999; Zbl 0958.22013)]). This simplifies the argument and gives at the same time a shorter proof for the known case of non-twisted endoscopy.

MSC:
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
22E99 Lie groups
Citations:
Zbl 0958.22013
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