Hecke modules and supersingular representations of U(2,1)

TL;DR

This paper classifies simple modules of a Hecke algebra associated with U(2,1), defines supersingular modules, and constructs supersingular representations of the group, advancing understanding of mod-p representations in p-adic groups.

Contribution

It provides a classification of simple modules for the pro-p-Iwahori-Hecke algebra of U(2,1) and establishes a correspondence between supersingular modules and supersingular representations.

Findings

Classification of simple modules for the Hecke algebra.

Definition of supersingular Hecke modules.

Construction of supersingular representations of U(2,1).

Abstract

Let F be a nonarchimedean local field of odd residual characteristic p. We classify finite-dimensional simple right modules for the pro-p-Iwahori-Hecke algebra $\mathcal{H}_C(G,I(1))$, where G is the unramified unitary group U(2,1)(E/F) in three variables. Using this description when C is the algebraic closure of $\mathbb{F}_p$, we define supersingular Hecke modules and show that the functor of I(1)-invariants induces a bijection between irreducible nonsupersingular mod-p representations of G and nonsupersingular simple right $\mathcal{H}_C(G,I(1))$-modules. We then use an argument of Paskunas to construct supersingular representations of G.