  Given a real irreducible dual pair there is an integral kernel operator which
maps the distribution character of an irreducible admissible representation of
the group with the smaller or equal rank to an invariant eigendistribution on
the group with the larger or equal rank. If the pair is in the stable range and
if the representation is unitary, then the resulting distribution is the
character of the representation obtained via Howe's correspondence. This
construction was transferred to the p-adic case and a conjecture was
formulated.
  In this note we verify a weaker version of this conjecture for dual pairs in
the stable range over a p-adic field.
