On the definition of transfer factors.(English)Zbl 0644.22005

“Endoscopic groups” in harmonic analysis over reductive groups $$G$$ on local fields $$F$$ are sophisticated objects and still elusive (an example is given by a 1-dimensional torus anisotropic over a local field $$F$$ of characteristic 0 but split over the quadratic extension of $$F$$, when $$G=\mathrm{SL}_2)$$. Such groups were introduced with the object of understanding the inner structure of “$$L$$-packets” of irreducible representations of the group $$G(F)$$ of $$F$$-valued points on $$G$$ (if $$\overline F$$ is the algebraic closure of F, conjugacy realized via elements of $$G(\overline F)$$ intervenes during the application of harmonic analysis to number- theoretic problems like the study of $$L$$-functions; for irreducible representations of $$G(F)$$, this conjugacy leads to a classification coarser than the usual equivalence and the coarse classes are called “$$L$$-packets” of irreducible representations). To a conjugacy class in the group $$H(F)$$ of $$F$$-rational points on an endoscopic group $$H$$ associated to $$G$$, there correspond many conjugacy classes in $$G(F)$$. “Orbital integrals” defined for pairs $$f$$, $$f^H$$ of functions respectively on $$G(F)$$, $$H(F)$$ are related by “transfer factors”. The object of this important paper is, inter alia, to define explicitly “transfer factors”, in the context of the correspondence of an $$f^H$$ on $$H(F)$$ to each $$f$$ on $$G(F)$$, with good functional behaviour: the definition, even for $$F=\mathbb R$$, represents an improvement upon an earlier construction.

MSC:

 22E50 Representations of Lie and linear algebraic groups over local fields 43A80 Analysis on other specific Lie groups 11F70 Representation-theoretic methods; automorphic representations over local and global fields
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