Tensor Products of Holomorphic Discrete Series Representations

Source abstract

We discuss the decomposition of tensor products of holomorphic discrete series representations, generalizing a technique used in [ 9 ] for representations of SL 2 ( R ), based on a suggestion of Roger Howe. In the case of two representations with highest weights, the discussion is entirely algebraic, and is best formulated in the context of generalized Verma modules (see § 3). In the case when one representation has a highest weight and the other a lowest weight, the approach is more analytic, relying on the realization of these representations on certain spaces of holomorphic functions. For a simple group, these two cases exhaust the possibilities; for a nonsimple group, one has to piece together representations on the various factors. The author wishes to thank Roger Howe and Jim Lepowsky for very helpful conversations, and Nolan Wallach for pointing out the work of Eugene Gutkin (Thesis, Brandeis University, 1978), from which some of the results of this paper can be read off as easy corollaries.