The wave front set and the asymptotic support for \(p\)-adic groups (Q1175080)

It is shown that for a \(p\)-adic reductive group \(G\) the wavefront set of the character \(\Theta_ \pi\) of an irreducible admissible representation \(\pi\) coincides with its asymptotic support. This is the \(p\)-adic analogue of a conjecture in the real case of \textit{D. Barbasch} and \textit{D. Vogan} [J. Funct. Anal. 37, 27-55 (1980; Zbl 0436.22011)]. It is possible to prove this since by results of Harish Chandra we not only have an asymptotic expansion of \(\Theta_ \pi\) by homogeneous distributions as in the real case [ibid.{]} but \(\Theta_ \pi\) already equals a finite sum of homogeneous distributions. From this the conclusion quickly follows.