On certain distinguished unitary representations supported on null cones

Let [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] = [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] or [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /], and let G = U(p, q; [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /]) be the isometry group of a [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="05i" /]-hermitian form of signature (p, q). For 2n ≤ min (p, q), we consider the action of G on Vⁿ, the direct sum of n copies of the standard module V = [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="06i" /], and the associated action of G on the regular part of the null cone, denoted by X⁰⁰. We show that there is a commuting set of G-invariant differential operators acting on the space of C^∞ functions on X⁰⁰ which transform according to a distinguished GL(n, [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="07i" /]) character, and the resulting kernel is an irreducible unitary representation of G. Our result can be interpreted as providing a geometric construction of the theta lift of the characters from the group G' = U(n, n) or O* (4n). The construction and approach here follow a previous work of Zhu and Huang [Representation Theory1 (1997)] where the group concerned is G = O(p, q) with p + q even.