Tamely ramified supercuspidal representations

Let G be a connected reductive group defined over a complete local non archimedean field F, and let G denote its F-valued points. Let (pi, V) be an irreducible admissible representation of G, and let (sigma, W) be a representation of a parahoric subgroup P which is trivial on the prounipotent radical U of P. We say that (pi, V) contains (sigma, W) if, when restricted to P, the sigma-isotypic part V-sigma is nontrivial. Assume that (sigma, W) is irreducible cuspidal on the finite group of Lie type P/U, and that V-sigma not equal 0. We show that (pi, V) is supercuspidal if and only if P is maximal; in this case pi is compactly induced from the normaliser of P. We then classify supercuspidal representations containing unipotent cuspidal representations, provided G is an inner form of a split adjoint group, following a conjecture of G. Lusztig.