Parameters and 2-cocycles in the representation theory of $p$-adic groups

In this talk, I will first explain how the category of smooth complex representations of a $p$-adic group can be described in terms of modules over a semidirect product of an affine Hecke algebra and a twisted group algebra. I will then discuss some recent results concerning the parameters of the affine Hecke algebras and the 2-cocycles defining the twisted group algebras. These results give a more explicit description of the algebras that arise in the representation theory of $p$-adic groups.