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

Abstract

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.

arXiv commentMinor updates after the referee's report

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• A family of irreducible supersingular representations of ${\mathrm{GL}}_2(F)$ for some ramified $p$-adic fields