Analytic families of eigenfunctions on a reductive symmetric space

In harmonic analysis on a reductive symmetric space X an important role is played by families of generalized eigenfunctions for the algebra D (X) of invariant dierential operators. Such families arise for instance as matrix coeÆcients of representations that come in series, such as the (generalized) principal series. In particular, relations between such families are of great interest. We recall that a real reductive group G; equipped with the left times right multiplication action, is a reductive symmetric space. In the case of the group, examples of the mentioned relations are functional equations for Eisenstein integrals, see [23] and [25], or Arthur-Campoli relations for Eisenstein integrals, see [1], [14]. In this paper we develop a general tool to establish relations of this kind. We show that they can be derived from similar relations satised by the family of functions obtained by taking one particular coeÆcient in a certain asymptotic expansion. Since the functions in the family so obtained are eigenfunctions on symmetric spaces of lower split rank, this yields a powerful inductive method; we call it induction of relations. In the case of the group, a closely related lifting theorem by Casselman was used by Arthur in the proof of the Paley-Wiener theorem, see [1], Thm. II.4.1. However, no proof seems yet to have appeared of Casselman's theorem