Cocenters and representations of pro-$p$ Hecke algebras

TL;DR

This paper explores the connection between cocenters and representations of affine pro-$p$ Hecke algebras, providing a new criterion to identify supersingular representations based on character vanishing.

Contribution

It introduces a novel criterion for supersingularity in representations, linking it to the behavior of characters on the cocenter's non-supersingular part.

Findings

Character vanishes on non-supersingular cocenter parts for supersingular representations

Provides a new characterization of supersingularity in affine pro-$p$ Hecke algebra representations

Enhances understanding of the structure of pro-$p$ Hecke algebra representations

Abstract

In this paper, we study the relation between the cocenter and the representations of an affine pro-$p$ Hecke algebra. As a consequence, we obtain a new criterion on the supersingular representation: a (virtual) representation is supersingular if and only if its character vanishes on the non-supersingular part of the cocenter.