Intégrales dʹEisenstein pour les espaces symétriques réductifs: Tempérance, majorations. Petite matriceB

Résumé majorations for the meromorphic continuation of Eisenstein integrals for reductive symmetric spaces are proved. Using a technique due to E. P. van den Ban, the problem reduces to prove temperedness when the parameter is pure imaginary. Starting with a result for “good” parameters, established by using a result of Flensted-Jensen, Oshima, and Schlichtkrull (1985) one gets temperedness by a repeated use of translations functors: for discrete series (cf. Vogan (1988)), for generalized principal series and for the Poisson transform. A key role is played by Oshimaʹs (1988) criterion for temperedness. The existence of a smallB-matrix is also proved.