We investigate the irreducible cuspidal C-representations of a reductive
p-adic group G over a field C of characteristic different from p. When
C is algebraically closed, for many groups G, a list of cuspidal C-types
(J,位) has been produced satisfying exhaustion, sometimes for a
restricted kind of cuspidal representations, and often unicity. We verify that
those lists verify Aut(C)-stability and we produce similar lists when C is
no longer assumed algebraically closed. Our other main results concern
supercuspidality. This notion makes sense for the representations 位 in
the cuspidal C-types (J,位) as above, which involve finite reductive
groups. We check that an irreducible cuspidal representation of G induced
from 位 is supercuspidal if and only 位 is supercuspidal.Comment: 45 pages This is the final versio