This talk is about a geometric construction of irreducible representations of groups like GL(n) and Sp(n) over finite fields, via Euler characteristics of algebraic varieties. It fits into a conceptual description of their representation theory which makes sense over any base field, and there’s some ongoing speculation that it could lift to characteristic zero and provide the “depth zero case” of the Langlands conjectures over local fields. In some cases, the theory allows for very explicit examples (we are talking “write down equations”); I’ll illustrate what’s going on focusing on GL(2).