  The wavefront set is a fundamental invariant arising from the
Harish-Chandra-Howe local character expansion of an admissible representation.
We prove a precise formula for the wavefront set of an irreducible
Iwahori-spherical representation with `real infinitesimal character' and
determine a lower bound for this invariant in terms of the
Deligne-Langlands-Lusztig parameters. In particular, for the Iwahori-spherical
representations with real infinitesimal character, we deduce that the algebraic
wavefront set is a singleton, as conjectured by Moeglin and Waldspurger. As a
corollary, we obtain an explicit description of the wavefront set of an
irreducible spherical representation with real Satake parameter.
