Abstract: The local Langlands program bridges between reductive groups over a local field F and data associated to the absolute Galois group of F are investigated. More specifically, the local Langlands correspondence matches irreducible representations of a reductive p-adic group G with Langlands parameters (valued in the complex dual group). Many notions for G-representations (e.g. tempered) have natural counterparts for L-parameters (e.g. bounded). In the modern version, with enhanced L-parameters, the Langlands correspondence becomes a bijection, so in principle any property of irreducible G-representations can be translated to a property of enhanced L-parameters. In this talk we will take a look at four well-known objects in the theory of reductive $p$-adicgroups: supercuspidal representations, the cuspidal support map, Bernstein components and Hecke algebras. For all four, we devise a good counterpart in the setting of the L-parameters. These counterparts do not rely on anything p-adic, they are defined entirely in terms of complex reductive groups with a Galois action. Thus we erect an entire new framework on L-parameters, analogous to but independent of the representation theory of reductive groups. We hope that this framework will be useful to establish new cases of the local Langlands correspondence. If time permits, we will also discuss how this setupenables us to relate representations of Hecke algebras to perverse sheaves on varieties of L-parameters. This talk is based on joint work with Anne-Marie Aubert and Ahmed Moussaoui.