Abstract The purpose of this paper is to study the local zeta integrals of Friedberg-Jacquet at complex place and to establish similar results to the recent work [4] joint with C. Chen and D. Jiang. In this paper, we will (1) give a necessary and sufficient condition on an irreducible essentially tempered cohomological representation π of GL 2 n ( C ) with a non-zero Shalika model; (2) construct a new twisted linear period Λ s , χ and give a different expression of the linear model for π; (3) give a necessary and sufficient condition on the character χ such that there exists a uniform cohomological test vector v ∈ V π (which we construct explicitly) for Λ s , χ . As a consequence, we obtain the non-vanishing of local Friedberg-Jacquet integral at complex place. All of the above are essential preparations for attacking a global period relation problem in the forthcoming paper ( [11] ).

中文翻译：

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GL2 的阿基米德非零、上同调测试向量和标准 L 函数：复杂情况

摘要 本文的目的是研究复数处Friedberg-Jacquet的局域zeta积分，并建立与C. Chen和D. Jiang最近的工作[4]类似的结果。在本文中，我们将 (1) 给出具有非零 Shalika 模型的 GL 2 n ( C ) 的不可约本质调和上同调表示 π 的充分必要条件；(2) 构造一个新的扭曲线性周期 Λ s , χ 并给出 π 的线性模型的不同表达式；(3) 给出字符 χ 的充分必要条件，使得对于 Λ s , χ 存在统一的上同调测试向量 v ∈ V π（我们明确构造）。因此，我们在复数处获得了局部 Friedberg-Jacquet 积分的非消失。