Let l be a prime. The generic l-modular representation and homotopy theory of finite reductive groups generalises to the setting of so-called "spetses" associated to l-adic reflection groups. These are "phantom" group-like objects about which we still have much "physical" information (classifying spaces, fusion systems, character values, etc.) built from the associated Hecke algebras and l-compact groups. I will report on the status of the Spetses programme, which is concerned with (a) finding and attaching this information, and (b) checking that it behaves as one might expect from the classical theory. These pursuits have led to the construction of several surprising bridges between different areas of mathematics.
