On certain distinguished unitary representations supported on null cones

Abstract

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 2 n ≤ min ( p, q ), we consider the action of G on V n , 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 X 00 . We show that there is a commuting set of G -invariant differential operators acting on the space of C ∞ functions on X 00 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 * (4 n ). 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.