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We study a certain family of holomorphic discrete series representations of the semisimple Lie group G = SU(n, n) and the corresponding analytic continuation in the inducing parameter X. At the values of X where the representations become reducible, we compute the composition series in terms of a Peter-Weyl basis on the Shilov boundary of the Herinitian symmetric space for G. Introduction. Holomorphic vector bundles E over a Hermitian symmetric space G/' K have been studied extensively, in particular in connection with realization of the holomorphic discrete series for G. One knows in certain cases that by considering the Harish-Chandra module M for G consisting of AT-finite sections of E, it is possible to obtain an analytic continuation in the parameters of the representation of K defining E, so that new (nondiscrete) unitary representations of G ensue [5]. These are bottom subquotients of M and defined by differential equations on the Shilov boundary. In this paper we will describe the full composition series for M in the special case of line bundles for the group SU(n, n) and its universal covering. Our computation of the composition series is not based (as usual) on intertwining operators (partly because the subquotients seem to be new, mostly nonunitary, representations) but rather on the behavior of the G-invariant (nondefinite in general) Hermitian form on M. The basic formula is an expansion of the distribution det(l — x)~x on U(n) in terms of the Peter-Weyl L2-basis. In §1 we derive this "binominal formula" affording a determination of the invariant Hermitian form on M, and in §2 we derive the composition series, expressed in terms of a certain ordering of the holomorphic dual of U(n). Finally in §3 we list a result and a conjecture relating the Ti-spectrum and the A/-spectrum of the modules. The author has benefited from conversations with H. P. Jakobsen, I. E. Segal, B. Speh (whose paper [8] gives a thorough treatment of the case of 51/(2, 2) as well as SO(n, 2)) and M. Vergne. 1. Let G denote the group SU(n, n) [1] for a fixed n and K its maximal compact subgroup which is isomorphic to (7(1) X SU(n) X SU(n). The character X*:e»^eTM (1) Received by the editors October 9, 1979. AMS (MOS) subject classifications (1970). Primary 22E45, 43A85; Secondary 05A10, 32H10.

Composition series for analytic continuations of holomorphic discrete series representations of 𝑆𝑈(𝑛,𝑛)