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 Q p . We state conjectures on the smooth representations of 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 GL n . When n = 2 and K is unramified, we prove several cases of our conjectures, including new finite length results.

Representations of $p$-adic groups over coefficient rings

“…plattice L defined on some O E as above, of infinite length reduction, and another one L ′ of finite length reduction. Take L = ind G K W Z ac p for a K-stable Z ac p -lattice W Z ac p of W and L ′ = V ∩ ind G Γ 1 Z ac p for a small enough discrete cocompact subgroup Γ of G. 16 cuspidal and supersingular will be defined later 17 ℓ is reserved for the primes different from p, think r = ℓ or p 18 a free Z ac r -submodule of scalar extension V to Q ac r 19 the ring OE is principal but not Z ac r . The definition bypasses this difficulty…”

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

“…These families are parametrized by choices of a collection of isomorphisms between one-dimensional F p -vector spaces. While the construction of [7] is not exhaustive, the supersingular representations considered there play an important role in ongoing investigations towards the local Langlands correspondence; see [6] and references therein.…”