  Let $p$ be a prime number and $K$ a finite extension of $\mathbb{Q}_p$. We
state conjectures on the smooth representations of $\mathrm{GL}_n(K)$ that
occur in spaces of mod $p$ automorphic forms (for compact unitary groups). In
particular, when $K$ is unramified, we conjecture that they are of finite
length and predict their internal structure (extensions, form of subquotients)
from the structure of a certain algebraic representation of $\mathrm{GL}_n$.
When $n=2$ and $K$ is unramified, we prove several cases of our conjectures,
including new finite length results.
