On the Bernstein--Zelevinsky classification in families

TL;DR

This paper investigates how admissible representations of p-adic GL_n vary within families, demonstrating the rigidity of ramified parts and exploring applications to GL_n-eigenvarieties.

Contribution

It provides new insights into the Bernstein--Zelevinsky classification's behavior in families and establishes the rigidity of ramified components.

Findings

Ramified parts of families are rigid.

Results applicable to GL_n-eigenvarieties.

Enhanced understanding of representation variation in p-adic groups.

Abstract

We study the variation of admissible representations of $p$-adic $GL_n$ in families from the point of view of the Bernstein--Zelevinsky classification and show that the ramified parts of these families are rigid. We explain how to apply our results in the context of $GL_n$-eigenvarieties.