On the definition of transfer factors
``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.
scientific article; zbMATH DE number 4049901