On some representations of the Iwahori subgroup

Let p⩾5 be a prime number. In [BL94] Barthel and Livné (1994) gave a classification for irreducible representations of GL2(F) over , for F a p-adic field, discovering some objects, referred to as “supersingular”, which appear as subquotients of universal representations . In this paper we give a detailed description of the Iwahori structure of such universal representations, in the case when F is an unramified extension of Qp. We determine a fractal structure which shows how and why the techniques used for Qp fail and which lets us determine“natural” subrepresentations of the universal object . As a corollary, we get the Iwahori structure of tamely ramified principal series.