Values of archimedean zeta integrals for unitary groups

Abstract

We prove that certain archimedean integrals arising in global zeta integrals involving holomorphic discrete series on unitary groups are predictable powers of π times rational or algebraic numbers. In some cases we can compute the integral exactly in terms of values of gamma functions, and it is plausible that the value in the most general case is given by the corresponding expression. Non-vanishing of the algebraic factor is readily demonstrated via the explicit expression. Regarding analytical aspects of such integrals, whether archimedean or p-adic, a recent systematic treatment is [Lapid–Rallis 2005] in the Rallis conference volume. In particular, the results of Lapid and Rallis allow us to focus on the arithmetic aspects of the special values of the integrals. This is implicit in (4.4)(iv) in [Harris 2006]. Roughly, the integrals here are those that arise in the so-called doubling method if the Siegel-type Eisenstein series is differentiated transversally before being restricted to the smaller group. In traditional settings, details involving Fourier expansions would be apparent, but such details are inessential. Specifically, many Fourier- expansion details concerning classical Maaß–Shimura operators are spurious, refer- ring, in fact, only to the structure of holomorphic discrete series representations. Of course, the translation to and from rationality issues in spaces of automorphic forms should not be taken lightly.