Let F / Q_p be a p-adic field.  In contrast to the situation for complex representations, no classification of the irreducible supercuspidal mod p representations of GL_n(F) is known, except in the case GL_2(Q_p).  If F / Q_p is unramified and r is a generic irreducible two-dimensional mod p representation of the absolute Galois group of F, then nearly 15 years ago Breuil and Paskunas gave a beautiful construction of an infinite family of diagrams giving rise to supercuspidal mod p representations of GL_2(F) with GL_2(O_F)-socle determined by Serre’s modularity conjecture for r.  While their construction is not exhaustive, various local-global compatibility results obtained by a number of mathematicians in the intervening years indicate that it is sufficiently general to capture the mod p local Langlands correspondence for generic Galois representations.

In this talk we will review the ideas mentioned above and discuss how to move beyond them to consider ramified p-adic fields F, or non-generic representations r for unramified F.  We will describe a simple construction of supercuspidal representations for certain ramified F and generic r; while this is the first such example for ramified F, it involves a breakage of symmetry that makes it unlikely to figure in the local Langlands correspondence for r.  We then discuss works in progress with Ariel Weiss and with Reem Waxman that shed new light on Breuil-Paskunas and aim to give a “correct” generalization of their construction.  A new feature is that we work with the category of mod p representations of GL_2(R), where R is a quotient ring of O_F that is larger than the residue field.