Archimedean Zeta Integrals for Unitary Groups

We derive precise formulas for the archimedean Euler factors occurring in certain standard Langlands $L$-functions for unitary groups. In the 1980s, Paul Garrett, as well as Ilya Piatetski-Shapiro and Stephen Rallis (independently of Garrett), discovered integral representations of automorphic $L$-functions that are Eulerian but, in contrast to the Rankin--Selberg and Langlands--Shahidi methods, do not require that the automorphic representations to which the $L$-functions are associated are globally generic. Their approach, the doubling method, opened the door to a variety of applications that could not be handled by prior methods. For over three decades, though, the integrals occurring in the Euler factors at archimedean places for unitary groups eluded precise computation, except under particular simplifications (such as requiring certain representations to be one-dimensional, as Garrett did in the first major progress on this computation and only prior progress for general signatures). We compute these integrals for holomorphic discrete series of general vector weights for unitary groups of any signature. This has consequences not only for special values of $L$-functions in the archimedean setting, but also for $p$-adic $L$-functions, where the corresponding term had remained open.