Finite length for unramified $\mathrm{GL}_2$

TL;DR

This paper proves that certain smooth representations of alg_2(K) arising in mod p cohomology are of finite length when p is sufficiently large, and provides new structural insights into these representations.

Contribution

It establishes finite length for admissible smooth alg_2(K) representations in mod p cohomology under generic conditions and large p, along with new structural results.

Findings

Representations have finite length under specified conditions.

New structural properties of alg_2(K) representations.

Results depend on p being large relative to [K:alq_p].

Abstract

Let $p$ be a prime number and $K$ a finite unramified extension of $\mathbb{Q}_p$. If $p$ is large enough with respect to $[K:\mathbb{Q}_p]$ and under mild genericity assumptions, we prove that the admissible smooth representations of $\mathrm{GL}_2(K)$ that occur in Hecke eigenspaces of the mod $p$ cohomology are of finite length. We also prove many new structural results about these representations of $\mathrm{GL}_2(K)$ and their subquotients.