If
is a quadratic extension
-adic fields, we first prove
that the
-distinguished
representations inside a distinguished unitary
-packet of
are precisely
those admitting a degenerate Whittaker model with respect to a degenerate character
of
.
Then we establish a global analogue of this result. For this, let
be a quadratic extension of number fields, and let
be an
-distinguished
square-integrable automorphic representation of
. Let
be the unique pair
associated to ,
where
is a cuspidal
representation of
with
.
Using an unfolding argument, we prove that an element of the
-packet of
is distinguished
with respect to
if and only if it has a degenerate Whittaker model for a degenerate character
of type
of
which is
trivial on ,
where
is the group of unipotent upper triangular matrices of
.
As a first application, under the assumptions that
splits at
infinity and
is odd, we establish a local–global principle for
-distinction
inside the
-packet
of
. As
a second application we construct examples of distinguished cuspidal automorphic
representations
of
such that the period integral vanishes on some canonical realization of
,
and of everywhere locally distinguished representations of
such that
their
-packets
do not contain any distinguished representation.