Conjectures and Results on Modular Representations of 𝐺𝐿𝑛(𝐾) for a 𝑝-Adic Field 𝐾

Let p p be a prime number and K K a finite extension of Q p Qₚ . We state conjectures on the smooth representations of G L n ( K ) GL_n(K) that occur in spaces of mod p p automorphic forms (for compact unitary groups). In particular, when K 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 G L n GL_n . When n = 2 n=2 and K K is unramified, we prove several cases of our conjectures, including new finite length results.