On certain distinguished unitary representations supported on null cones

Abstract

Let double-struck F = ℂ or ℍ, and let G = U(p, q; double-struck F) be the isometry group of a double-struck F-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 = double-struck Fp+q, 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, double-struck F) 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 Theory 1 (1997)] where the group concerned is G = O(p, q) with p + q even.