Local character expansions and asymptotic cones over finite fields

We generalise Gelfand-Graev characters to $\mathbb{R}\mathrm{/}\mathbb{Z}$-graded Lie algebras and lift them to produce new test functions to probe the local character expansion in positive depth. We show that these test functions are well adapted to compute the leading terms of the local character expansion and relate their determination to the asymptotic cone of elements in $\mathbb{Z}\mathrm{/}n$-graded Lie algebras. As an illustration, we compute the geometric wave front set of certain toral supercuspidal representations in a straightforward manner.