论文标题
统一团体的Archimedean Zeta积分
Archimedean Zeta Integrals for Unitary Groups
论文作者
论文摘要
我们为在某些标准Langlands $ l $ unctions中为单一群体提供的Archimedean Euler因子提供了精确的公式。 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是全球通用的。他们的方法是加倍方法，为无法通过先前方法来处理的各种应用程序打开了大门。 但是，在过去的三十年中，除了特定的简化（例如，要求某些代表是一维的，就像Garrett在此计算上一样，只有一维的主要进展，仅在特定的简化下，仅在特定的简化中出现了统一组的欧拉因子中的积分。我们计算了这些签名单一组的全体形态离散系列的一系列常规矢量权重系列的积分。这不仅是对阿基赛马环境中$ l $ functions的特殊值的后果，而且还对$ p $ - adic $ l $ functions，相应的期限保持开放。
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.
