The pro-p-Iwahori invariants of the universal quotient representation for GL2(F)

Abstract

Let [Formula: see text] be a finite extension of [Formula: see text]. Every supersingular mod [Formula: see text] representation of [Formula: see text] is a quotient of a certain universal module, say [Formula: see text] The space of pro-[Formula: see text]-Iwahori invariants of [Formula: see text] is instrumental in classifying the supersingular mod [Formula: see text] representations of [Formula: see text]. For [Formula: see text] the space of pro-[Formula: see text]-Iwahori invariants of [Formula: see text] has been determined by Schein and Hendel using the spherical Hecke algebra. An alternative way to study mod [Formula: see text] representations of [Formula: see text] is introduced by Anandavardhanan and Borisagar using the Iwahori–Hecke algebra. In this paper, we give a canonical isomorphism between the universal modules for [Formula: see text] obtained from the spherical approach and the Iwahori–Hecke approach. We also compute an explicit basis of the pro-[Formula: see text]-Iwahori invariant space of [Formula: see text] for [Formula: see text] using the Iwahori–Hecke approach. This method allows us to extend Hendel’s computation slightly in the range of [Formula: see text]-adic digits from [Formula: see text] to [Formula: see text]