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.


Additional Information

Print ISSN
pp. 1059-1076
Launched on MUSE
Open Access
Back To Top

This website uses cookies to ensure you get the best experience on our website. Without cookies your experience may not be seamless.