Let G be a reductive p-adic group and let Rep(G)s Bernstein block in the category of smooth complex G-representations. We investigate structure Rep(G)s, by analysing algebra G-endomorphisms progenerator Π that category. show is “almost” Morita equivalent with (twisted) affine Hecke algebra. This statement made precise several ways, most importantly family graded algebras. It entails that, as fa...