On Paley-Wiener's theorem on Arthur

The Fourier transform of a C-infinity function, f, with compact support on a real reductive Lie group G is given by a collection of operators phi(P, sigma, lambda) := pi(P)(sigma, lambda) (f) for a suitable family of representations of G, which depends on a family, indexed by P in a finite set of parabolic subgroups of G, of pairs of parameters (sigma, lambda), sigma varying in a set of discrete series, lambda lying in a complex finite dimensional vector space. The pi(P)(sigma, lambda) are generalized principal series, induced from P. It is easy to verify the holomorphy of the Fourier transform in the complex parameters. Also it satisfies some growth properties. Moreover an intertwining operator between two representations pi(P)(sigma, lambda), pi(P')(sigma', lambda') of the family, implies an intertwining property for phi(P, sigma, lambda) and phi(P, sigma', lambda'). There is also a way to introduce "successive (partial) derivatives" of the family of representations, pi(P)(sigma, lambda), along the parameter lambda, and intertwining operators between subquotients of these successive derivatives imply the intertwining property for the successive derivatives of the Fourier transform phi. We show that these properties characterize the collections of operators (P, sigma, lambda) -> phi(P, sigma, lambda) which are Fourier transforms of a C-infinity function with compact support, for G linear. The proof, which uses Harish- Chandra's Plancherel formula, rests on a similar result for left and right K-finite functions, which is due to J. Arthur. We give also a proof of Arthur's result, purely in term of representations, involving the work of A. Knapp and E. Stein on intertwining integrals and Langlands and Vogan's classifications of irreducible representations of G.