We investigate the irreducible cuspidal -representations of a reductive -adic group over a field of characteristic different from . In all known cases, such a representation is the compactly induced representation from a smooth -representation of a compact modulo centre subgroup of . When is algebraically closed, for many groups , a list of pairs has been produced, such that any irreducible cuspidal -representation of has the form , for a pair unique up to conjugation. We verify that those lists are stable under the action of field automorphisms of , and we produce similar lists when is no longer assumed algebraically closed. Our other main result concerns supercuspidality. This notion makes sense for the irreducible cuspidal -representations of , but also for the representations above, which involve representations of finite reductive groups. In most cases we prove that is supercuspidal if and only if is supercuspidal.
"Representations of a reductive -adic group in characteristic distinct from ." Tunisian J. Math. 4 (2) 249 - 305, 2022. https://doi.org/10.2140/tunis.2022.4.249