One of the first basic ideas we all learn is that a continuous function is determined by its values on a dense open subset. In representation theory, this allows us to recognize a representation of a Lie group from an especially well-behaved locus--that of regular semisimple elements. But what if we want to study representations of matrix groups over finite fields? Lusztig's revolutionary idea in the 1980s was to replace the analytic notion of limit in the continuous setting by the algebro-geometric notion of intermediate extension in this discrete setting. I will explain this picture and describe a recent construction of perverse sheaves that give rise to positive-depth supercuspidal representations of p-adic groups. In the simplest nontrivial case, this resolves a 2004 conjecture of Lusztig. This is joint work with R. Bezrukavnikov.