Abstract
In this paper we consider the problem of decomposing tensor products of certain singular unitary representations of a semisimple Lie group G. Using explicit models for these representations (constructed earlier by one of us) we show that the decomposition is controlled by a reductive homogeneous space
$ G^{\prime}/H^{\prime} $
. Our procedure establishes a correspondence between certain unitary representations of G and those of
$ G^{\prime} $
. This extends the usual
$ \theta $
-correspondence for dual reductive pairs. As a special case we obtain a correspondence between certain representations of real forms of E
7 and F
4.