In this paper we study the
theta correspondence for Unitary groups of the same size over local and global fields.
This correspondence has been studied in many cases by several authors. We are able
to unify and generalise all these known results in terms of two conjectures, one local
and the other global. These conjectures are in terms of the parametrisation of
irreducible admissible representations of groups over local fields which are
formulated by David Vogan refining Langlands parametrization, and which are
now called Vogan parameters. In turn, the simple form of the conjecture
here, gives support to the importance of Vogan’s refinement of Langlands
parametrisation.
