Abstract: In the 1990s, Henniart proved that certain supercuspidal representations of p-adic GLn are characterized by their character values on very regular elements, a special class of regular semisimple elements on which character formulae are remarkably simple. Henniart’s result has seen many interesting applications—for example, in determining algebraic descriptions of geometrically arising representations. In this talk, we’ll discuss a generalization of Henniart’s theorem to general G. As a byproduct of our methods, we obtain an easy, non-cohomological condition distinguishing unipotent supercuspidal representations, yielding a p-adic analogue of Lusztig’s criterion for finite fields. This is joint work with M. Oi.