Analytic families of eigenfunctions on a reductive symmetric space

Let $X=G/H$ be a reductive symmetric space, and let $\mathbb D(X)$ denote the algebra of $G$-invariant differential operators on $X$. The asymptotic behavior of certain families $f_\lambda$ of generalized eigenfunctions for $\mathbb D(X)$ is studied. The family parameter $\lambda$ 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 theorem.