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We define a certain Hecke theoretic notion of family of smooth admissiblerepresentations of $GL_n(F)$, or of products of such groups, where $F$ is anonarchimedean local field of characteristic zero. While this notion of familyis rather weak a priori, we show that it implies strong rigidity properties forthe Bernstein-Zelevinsky presentations of the members of such families, as wellas for the variation of their epsilon factors. Examples of such families comeoften from the theory of eigenvarieties, and in this case our results implyanalyticity properties for the epsilon factors of the members of thesefamilies.

On the Bernstein-Zelevinsky classification and epsilon factors in families