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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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