Tensor products of singular representations and an extension of the θ-correspondence

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′/H′. Our procedure establishes a correspondence between certain unitary representations of G and those of G′. This extends the usual θ-correspondence for dual reductive pairs. As a special case we obtain a correspondence between certain representations of real forms of E₇ and F₄.