WAVE-PACKETS IN SCHWARTZ SPACE OF A REDUC TIVE SYMMETRICAL SPACE

We study holomorphic Families of K-finite eigenfunctions on symmetric spaces G/H, called functions IIhol(Lambda) by analogy with [HC]. Eisen stein integrals (cf. [B3], [D]), suitably normalized by a polynomial f actor, provide examples of such families. A function IIhol(Lambda) is said IIhol'(Lambda), if, roughly speaking, its constant term along any sigma theta-stable parabolic subgroup is a finite sum of functions II hol(Lambda(s)), where Lambda(s) varies in a determined finite set. We prove that, for a function IIhol'(Lambda), one can form wave packets i n the Schwartz space. We prove also a criterion for a function IIhol(L ambda) to be IIhol'(Lambda). An important fact is that, for minimal si gma theta-stable parabolic subgroups, our criterion implies, with the help of the Maas-Selberg relations (cf. [B2], [B3]), a normalization o f Eisenstein integrals. All the article relies on the theory of the co nstant term (cf. [C]). (C) 1996 Academic Press, Inc.