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.

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