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Foundations of twisted endoscopy. (English) Zbl 0958.22013
Astérisque. 255. Paris: Société Mathématique de France, 190 p. (1999).
The theory of twisted endoscopy is an extension of standard endoscopy to the following situation. Let $$G$$ be a connected reductive group over a local or global field $$F$$ of characteristic zero. Let $$\theta$$ be an automorphism of $$G$$ over $$F$$ and let $$\omega$$ be a one-dimensional representation of $$G(F)$$ (resp. of $$G({\mathbb A})$$, trivial on $$G(F)$$). Twisted endoscopy then is related to representations $$\pi$$ of $$G(F)$$ (resp. $$G({\mathbb A})$$) such that $$\pi\circ\theta$$ is equivalent to $$\omega\otimes\pi$$. It serves, in particular, in the stabilization of the $$(\theta,\omega)$$-twisted trace formula, as standard endoscopy does for the ordinary trace formula. A non-initiated reader is recommended to read section 3 of J. Arthur [Unipotent automorphic representations: Conjectures. Astérisque 171-172, 13-71 (1989; Zbl 0728.22014)], where the notions around standard endoscopy are briefly discussed. In the present book the authors give the general definition of endoscopic data. They define a norm-correspondence between twisted semisimple conjugacy classes in $$G$$ and semisimple conjugacy classes in an endoscopic group. Next the local transfer factors are defined and their properties established. Then the matching of functions is defined and the stabilization of the elliptic part of the twisted trace formula is obtained.

##### MSC:
 22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings 11F72 Spectral theory; trace formulas (e.g., that of Selberg) 22E50 Representations of Lie and linear algebraic groups over local fields 11R34 Galois cohomology 22-02 Research exposition (monographs, survey articles) pertaining to topological groups