Mod p representations of GL(2,F)

Abstract:
Mod $p$ representation theory of $p$-adic reductive groups originated with the work of Barthel and Livne from the mid nineties where they studied irreducible smooth mod $p$ representations of $GL(2,F)$. Though there has been a lot of recent progress, a complete classification of all smooth mod $p$ representations of $G(F)$ is achieved only in very few cases, essentially when $G=GL(2)$ or one of its variants and when $F=Q_p$. The case of $GL(2,Q_p)$ follows from the works of Barthel-Livne and Breuil. In this talk we will survey the case of $GL(2,Q_p)$ and indicate an approach to study this question for $GL(2,F)$ where $F$ is an unramified extension of $Q_p$. This latter work is joint with Arindam Jana.