Tensor Products , Reproducing Kernels , and Power Series

Let 9 be a connected complex domain and let G be a group of holomorphic transformations of 9. Let Hi (i = 1,2) be reproducing kernel Hilbert spaces of holomorphic functions from 9 to finite dimensional complex vector spaces Vi such that each Hi carries a unitary representation Vi of G. Then U1 @ Us is unitary in a reproducing kernel Hilbert space of holomorphic functions from 5-3 X 9 to Vi @ V, . Since 3 sits naturally in 9 X 5B, one can attempt to decompose lJ, @ Us by restricting it to 9. Clearly this will not give the complete decomposition, since a function can vanish on B without being zero on 9 x 9. One is then lead to consider “derivatives perpendicular to the diagonal.” We make this notion precise for G = SU(n, n) and G = Sp(n, R), and show that one in fact can get the decomposition this way. In Section 1 we give some key relations obtainable from quite general properties of reproducing kernels. In Section 2 we use these to find composition series for modules of holomorphic functions. Finally, in Section 3, we introduce the Hilbert space structures and treat the case of tensor products of holomorphic discrete series. This is illustrated by some examples with G = SU(2,2). We acknowledge that the results of Michele Vergne [ll] have inspired this work. We are also indebted to A. Mayer and I. E. Segal for friendly help and conversations.