  In this paper we extend Barbasch and Moy's results about the local character
expansion of depth-$0$ representations to arbitrary depth. This leads to the
definition of a graded linearized version of the Kawanaka wavefront set which
we use to prove that for unramified groups, the toral supercuspidal
representations attached to unramified tori have singleton geometric wavefront
sets provided the residue characteristic is greater than $3(h-1)$ where $h$ is
the Coxeter number for $\mathbb G$. In the process we introduce the notion of
the asymptotic cone in the context of geometric invariant theory and prove it
is irreducible in the case of a reductive group acting on its Lie algebra via
the adjoint action using Borho's theory of $\mathbb G$-sheets.
