Cocenters and representations of pro-$p$ Hecke algebras
  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. 
Xuhua He and Sian Nie
math.RT