Tensor Products of Analytic Continuations of Holomorphic Discrete Series

We give the irreducible decomposition of the tensor product of an an- alytic continuation of the holomorphic discrete series of SU(2, 2) with its conjugate. 0. Introduction. The work of Segal (IES) and Mautner (M) established the abstract Plancherel theorem for type I groups. This meant that for an arbitrary unitary represen- tation, one could find its spectral decomposition into irreducibles and a corresponding spectral measure. To make this program explicit on L 2 -spaces on homogeneous spaces is one of the main subjects of harmonic analysis. Another interesting case is that of decom- posing a tensor product of irreducible representations; our aim in this paper is to consider this for certain holomorphic representations. The problem of finding the irreducible decomposition of tensor products of holomor- phic discrete series of the group SL(2, ) has been studied by Repka (Re1). The results there were used by Howe (How) to give the decomposition of the metaplectic represen- tation for certain dual pairs. See also (OZ). For a general semisimple Lie group G of Hermitian type a similar problem is studied in (Re2). It is shown that the tensor prod- uct of a scalar holomorphic discrete series with its conjugate is unitarily equivalent to the L 2 -space on the corresponding Hermitian symmetric space, L 2 (G K). Therefore we know its decomposition from the known theory of Harish-Chandra; namely L 2 (G K) W H ( ) C( ) 2 d