Tensor products of singular representations and an extension of the $ \theta $-correspondence

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.