emile.okada@ntu.edu.sgThe character of a representation of a p-adic group tells us a great deal about the representation. Remarkably, near the identity, its singularities can be described by a finite sum of Fourier transforms of nilpotent orbital integrals. This local character expansion connects representation theory with the geometry of nilpotent orbits. The leading orbits form the representation’s wavefront set, an invariant that links character singularities, the growth of fixed-vector spaces, and Whittaker models.
In this talk, I will present joint work with Dan Ciubotaru on computing wavefront sets for representations of positive depth. We use carefully constructed test functions to reduce the problem to questions about rational nilpotent degenerations in Lie algebras over finite fields. I will explain how asymptotic cones enter the picture and describe our results for representations of integral depth. Along the way, I will use explicit character formulae to motivate the wavefront set and explain its arithmetic significance.