ON CERTAIN DISTINGUISHED UNITARY REPRESENTATIONS SUPPORTED ON NULL CONES

Let F = C or H, and let G = U(p, q; F) be the isometry group of a F-he rmitian form of signature (p, q). For 2n less than or equal to min (p, q), we consider the action of G on V-n, the direct sum of n copies of the standard module V = Fp+q, and the associated action of G on the r egular part of the null cone, denoted by chi(00). We show that there i s a commuting set of G-invariant differential operators acting on the space of C-infinity functions on chi(00) which transform according to a distinguished GL(n, F) character, and the resulting kernel is an irr educible unitary representation of G. Our result can be interpreted as providing a geometric construction of the theta lift of the character s from the group G' = U(n, n) or O(4n). The construction and approach here follow a previous work of Zhu and Huang [Representation Theory 1 (1997)] where the group concerned is G = O(p, q) with p + q even.