Ec. Tan et Cb. Zhu, ON CERTAIN DISTINGUISHED UNITARY REPRESENTATIONS SUPPORTED ON NULL CONES, American journal of mathematics, 120(5), 1998, pp. 1059-1076
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.