  We define a certain Hecke theoretic notion of family of smooth admissible
representations of $GL_n(F)$, or of products of such groups, where $F$ is a
nonarchimedean local field of characteristic zero. While this notion of family
is rather weak a priori, we show that it implies strong rigidity properties for
the Bernstein-Zelevinsky presentations of the members of such families, as well
as for the variation of their epsilon factors (attached to arbitrary functorial
lifts). Examples of such families come often from the theory of eigenvarieties,
and in this case our results imply analyticity properties for the epsilon
factors of the members of these families.
