Homological duality for covering groups of reductive $p$-adic groups

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). Along way also study Grothendieck--Serre with respect Bernstein center provide proof folklore result on admissible modules nothing but contragredient duality. We out necessary sufficient condition when these three dualities agree length given block. particular, show all cuspidal blocks as well as, due Roche, trivial stabilizer relative Weyl