In this largely expository paper we extend properties of the homological duality functor $RHom_{\mathcal H}(-,{\mathcal H})$ where ${\mathcal H}$ is Hecke algebra a reductive $p$-adic group, to case it finite central extension group. The most important being that concentrated in single degree for irreducible representations and gives rise Schneider--Stuhler Ext groups (a Serre like property). A...