Representations of a reductive p-adic group in characteristic distinct from p

We investigate the irreducible cuspidal C-representations of a reductive p-adic group G over a field C of characteristic different from p. In all known cases, such a representation is the compactly induced representation indGJ lambda from a smooth C-representation lambda of a compact modulo centre subgroup J of G. When C is algebraically closed, for many groups G, a list of pairs (J, lambda.) has been produced, such that any irreducible cuspidal C-representation of G has the form indGJ lambda , for a pair (J,lambda) unique up to conjugation. We verify that those lists are stable under the action of field automorphisms of C, and we produce similar lists when C is no longer assumed algebraically closed. Our other main result concerns supercuspidality. This notion makes sense for the irreducible cuspidal C-representations of G, but also for the representations lambda above, which involve representations of finite reductive groups. In most cases we prove that indGJ lambda is supercuspidal if and only if lambda. is supercuspidal.