On the finite length of some $p$-adic representations of the quaternion algebra over $\Q_p$

TL;DR

This paper proves that certain admissible unitary Banach space representations of the non-split quaternion algebra over p are topologically of finite length, contributing to the understanding of their structural properties.

Contribution

It establishes the finite length property for a class of admissible unitary Banach space representations of the quaternion algebra over p, a novel result in this area.

Findings

Certain admissible unitary Banach space representations are of finite length.

The result applies specifically to the non-split quaternion algebra over p.

This advances the structural understanding of these representations.

Abstract

Let $D$ be the non-split quaternion algebra over $\Q_p$. We prove that a class of admissible unitary Banach space representations of $D^{\times}$ are topologically of finite length.