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Let $p$ be a prime number and $K$ a finite extension of $\mathbb{Q}_p$. Westate conjectures on the smooth representations of $\mathrm{GL}_n(K)$ thatoccur in spaces of mod $p$ automorphic forms (for compact unitary groups). Inparticular, when $K$ is unramified, we conjecture that they are of finitelength 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.

Conjectures and results on modular representations of $\mathrm{GL}_n(K)$ for a $p$-adic field $K$