Iwahori–Hecke model for the universal supersingular representation

Let F be a nonarchimedean local field with residue field ‫ކ‬ q .Every supersingular representation of GL 2 (F) is a quotient of a universal supersingular representation π r for some integer 0 ≤ r ≤ q -1.When F is a finite extension of ‫ޑ‬ p , Anandavardhanan-Borisagar and Anandavardhanan-Jana introduced an Iwahori-Hecke model for π r in the regular case 0 < r < q -1.When F = ‫ޑ‬ p , Chitrao introduced an Iwahori-Hecke model for π r when r = 0, p-1.This model was then used by Chitrao-Ghate to compute the reduction mod p of two-dimensional irreducible semistable representations of Gal(‫ޑ‬ p ‫ޑ/‬ p ).We extend this model to an arbitrary local field F for r = 0, q -1.We also write down an explicit nonsplit self-extension of π r which has a four-dimensional space of I (1)-invariants when F = ‫ޑ‬ p and r = 0, p -1.