  In analogy with the Barbasch-Vogan duality for real reductive linear groups,
we introduce a duality notion useful for the representation theory of the real
metaplectic groups. This is a map on the set of nilpotent orbits in a complex
symplectic Lie algebra, whose range consists of the so-called metaplectic
special nilpotent orbits. We relate this duality notion with the theory of
primitive ideals and extend the notion of special unipotent representations to
the real metaplectic groups. We also interpret the duality map in terms of
double cells of Weyl group representations.
