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تسجيل مستخدم جديدArchimedean Non-vanishing, Cohomological Test Vectors, and Standard $L$-functions of $mathrm{GL}_{2n}$: Complex Case
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The purpose of this paper is to study the local zeta integrals of Friedberg-Jacquet at complex place and to establish similar results to our recent work in the reall case joint with C. Cheng and D. Jiang. In this paper, we will (1) give a necessary and sufficient condition on an irreducible essentially tempered cohomological representation $pi$ of $mathrm{GL}_{2n}(mathbb{C})$ with a non-zero Shalika model; (2) construct a new twisted linear period $Lambda_{s,chi}$; (3) give a necessary and sufficient condition on the character $chi$ such that there exists a uniform cohomological test vector $vin V_pi$ (which we construct explicitly) for $Lambda_{s,chi}$. 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.
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The standard $L$-functions of $mathrm{GL}_{2n}$ expressed in terms of the Friedberg-Jacquet global zeta integrals have better structure for arithmetic applications, due to the relation of the linear periods with the modular symbols. The most technica
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