Prehomogeneous vector spaces and their zeta functions have been studied since the 1970s.
They occur naturally in the analysis of nilpotent orbits of semisimple Lie algebras and should play a central role in the finer analysis of Arthur's trace formula.
So far this has been carried out only in very special cases.
I will describe some new qualitative results which can be considered as a first step towards the general case.
No prior knowledge of the subject will be assumed.
Joint work with Tobias Finis