Symmetric Tube Domain

Let f! = G/K be a symmetric tube domain where G is the universal cover of Aut(f!). Let X be a line bundle on the Shilov boundary, and let I(x) be the space of sections. This paper determines (a) the composition series for I(x) as a ({1, K)module, (b) the K-module structure of each constituent, (c) explicit formulas for possible invariant Hermitian forms on these constituents, and (d) the unitarizable constituents. Introduction. Let n = G I K be a symmetric tube domain of rank n and let G be the universal covering group of Aut(O). The Shilov boundary of n is of the form G I P, where P = LN is a maximal parabolic subgroup with abelian nilradical N. Let x be a character of L such that the induced representation I(x) = Ind~(x) has a non-trivial, invariant, Hermitian form. This paper determines the composition series for such I(x), describes the K-module structure of each constituent, and obtains explicit formulas for the invariant Hermitian forms on the constituents. In particular, this leads to a complete determination of the unitarizable constituents of the I(x). The main idea is the following: After suitable normalization, the Hermitian form has a rational dependence on the parameter. Moreover, if x is suitably integral, then the Hermitian form is given by an equivariant differential operator and, as shown in [S], the Capelli identity of [KS] gives an explicit formula for this form at these points. In view of the rationality, this allows one to calculate Hermitian forms for all x, and everything else follows. This technique should perhaps be considered an "algebraic" continuation of the Capelli identity. 1991 Mathematics Subject Classification. Primary 22E46; Secondary 32M15; 53C35. This work was supported in part by an NSF grant at Princeton University. The author would also like to thank the Mehta Research Institute, Allahabad, India for its hospitality while this paper was being written. 275 © 1993 American Mathematical Society 0271-4132/93 Sl.OO + $.25 per page http://dx.doi.org/10.1090/conm/145/1216195 Licensed to Princeton University. Prepared on Sat Dec 15 18:51:55 EST 2012 for download from IP 128.112.200.107. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms