Vol. 315, No. 2, 2021

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Iwahori–Hecke model for mod $p$ representations of $\mathrm{GL}(2,F)$

U. K. Anandavardhanan and Arindam Jana

Vol. 315 (2021), No. 2, 255–283
DOI: 10.2140/pjm.2021.315.255

For a p-adic field F, the space of pro-p-Iwahori invariants of a universal supersingular mod p representation τ of GL2(F) is determined in the works of Breuil, Schein, and Hendel. The representation τ is introduced by Barthel and Livné and is defined in terms of the spherical Hecke operator. In Anandavardhanan and Borisagar [2013; 2015], an Iwahori–Hecke approach was introduced to study these universal supersingular representations in which they can be characterized via the Iwahori–Hecke operators. In this paper, we construct a certain quotient π of τ, making use of the Iwahori–Hecke operators. When F is not totally ramified over p, the representation π is a nontrivial quotient of τ. We determine a basis for the space of invariants of π under the pro-p-Iwahori subgroup. A pleasant feature of this “new” representation π is that its space of pro-p-Iwahori invariants admits a more uniform description vis-à-vis the description of the space of pro-p-Iwahori invariants of τ.

modular representations, supersingular representations, Iwahori–Hecke model
Mathematical Subject Classification
Primary: 20G05
Secondary: 11F70, 22E50
Received: 21 February 2021
Revised: 7 September 2021
Accepted: 1 November 2021
Published: 19 January 2022
U. K. Anandavardhanan
Department of Mathematics
Indian Institute of Technology Bombay
Arindam Jana
School of Mathematics
Tata Institute of Fundamental Research
Department of Mathematics
Indian Institute of Technology Bombay