## 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:

 2.2e+46 Representations of Lie and linear algebraic groups over real fields: analytic methods 2.2e+100 Lie groups

### Keywords:

endoscopic transfer; real Lie groups; orbital integrals

Zbl 0958.22013
Full Text:

### References:

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