Title: Wave-front sets and graded Springer theory

Abstract: For a character of a p-adic reductive group there is the notion of wave-front set, which is a set of nilpotent orbits that describes the asymptotic behavior of the character near the identity. By a theorem of Moeglin-Waldspurger, it also describes the least degenerate Whittaker models, which is a double generalization of local components of Fourier expansions for modular forms.

There is the conjecture that any wave-front set is contained in a single geometric orbit. This conjecture is confirmed for many cases of depth-0 representations, based on Lusztig's work on the analogous wave-front set question over the residue field. In this talk, we explain how the above conjecture does not hold in general, in particular not for a depth-1/2 representation we will construct, because the analogous conjecture does not hold for graded Lie algebras. This last observation is inspired by Springer theory for graded Lie algebras, which we hope to briefly talk about.