Analytic families of eigenfunctions on a reductive symmetric space. (English summary) Represent. Theory 5 (2001), 615–712 (electronic). Summary: “Let X = G/H be a reductive symmetric space, and let D(X) denote the algebra of G-invariant differential operators on X. The asymptotic behavior of certain families fλ of generalized eigenfunctions for D(X) is studied. The family parameter λ is a complex character on the split component of a parabolic subgroup. It is shown that the family is uniquely determined by the coefficient of a particular exponent in the expansion. This property is used to obtain a method by means of which linear relations among partial Eisenstein integrals can be deduced from similar relations on parabolic subgroups. In the special case of a semisimple Lie group considered as a symmetric space, this result is closely related to a lifting principle introduced by Casselman. The induction of relations will be applied in forthcoming work on the Plancherel and the Paley-Wiener theorems.