Cocenters and representations of pro-𝑝 Hecke algebras

Abstract

In this paper, we study the relation between the cocenter H ~ ¯ \overline {{\tilde {\mathcal H}}} and the representations of an affine pro- p p Hecke algebra H ~ = H ~ ( 0 , − ) {\tilde {\mathcal H}}={\tilde {\mathcal H}}(0, -) . As a consequence, we obtain a new criterion on supersingular representations: a (virtual) representation of H ~ {\tilde {\mathcal H}} is supersingular if and only if its character vanishes on the non-supersingular part of the cocenter H ~ ¯ \overline {\tilde {\mathcal H}} .