TAILIEUCHUNG - Đề tài " Sur le th´eor`eme de Paley-Wiener d’Arthur "

The Fourier transform of a C ∞ function, f , with compact support on a real reductive Lie group G is given by a collection of operators φ(P, σ, λ) := π P (σ, λ)(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 (σ, λ), σ varying in a set of discrete series, λ lying in a complex finite dimensional vector space. The π P (σ, λ) are generalized principal series, induced from P . It is. | Annals of Mathematics Sur le th eor eme de Paley-Wiener d Arthur By Patrick Delorme Annals of Mathematics 162 2005 987-1029 Sur le theoreme de Paley-Wiener d Arthur By Patrick Delorme Abstract The Fourier transform of a C function f with compact support on a real reductive Lie group G is given by a collection of operators Ộ P ơ A nP ơ A 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 ơ A ơ varying in a set of discrete series A lying in a complex finite dimensional vector space. The nP ơ A 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 nP ơ A nP ơ A of the family implies an intertwining property for ộ P ơ A and Ộ P ơ A . There is also a way to introduce successive partial derivatives of the family of representations nP ơ A along the parameter A and intertwining operators between subquotients of these successive derivatives imply the intertwining property for the successive derivatives of the Fourier transform Ộ. We show that these properties characterize the collections of operators P ơ A Ộ P ơ A which are Fourier transforms of a C x 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. 0. Introduction Le Theoreme de Paley-Wiener de J. Arthur cf. A decrit la transformee de Fourier de l espace des fonctions C a support compact T-spheriques ou K-finies a gauche et a droite sur un groupe reductif reel dans la classe d .

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