Let
be a
quadratic extension of nonarchimedean locally compact fields of residual characteristic
and let
denote its nontrivial
automorphism. Let
be an algebraically closed field of characteristic different from
. To any cuspidal
representation
of
, with coefficients
in
, such that
(such a representation is said
to be
-selfdual) we associate
a quadratic extension
,
where
is a tamely
ramified extension of
and
is a tamely ramified
extension of
, together with
a quadratic character of
.
When
is
supercuspidal, we give a necessary and sufficient condition, in terms of these data, for
to be
-distinguished. When
the characteristic
of
is not
, denoting by
the nontrivial
-character of
trivial on
-norms, we prove that any
-selfdual supercuspidal
-representation is either
distinguished or
-distinguished,
but not both. In the modular case, that is when
, we give examples
of
-selfdual
cuspidal nonsupercuspidal representations which are not distinguished nor
-distinguished. In the
particular case where
is the field of complex numbers, in which case all cuspidal representations are supercuspidal,
this gives a complete distinction criterion for arbitrary complex cuspidal representations,
as well as a purely local proof, for cuspidal representations, of the dichotomy and
disjunction theorem due to Kable and Anandavardhanan, Kable and Tandon,
when .
