Admissibility of Bernstein centers

Abstract

We provide a criterion for determining when elements of the Bernstein center of a totally disconnected locally compact group are admissible invariant distributions in the sense of Harish-Chandra \cite{HC99}. As a consequence, we deduce the local integrability results for elements of bounded depth or Bernstein supports in the Bernstein centers of reductive $p$-adic groups. Our methods apply uniformly to both complex and mod-$\ell$ coefficients, generalizing results of Moy and Tadić \cite{MT02} in the complex case.