Archimedean non-vanishing, cohomological test vectors, and standard L-functions of GL2n: Complex case

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 GL2n(C) with a non-zero Shalika model; (2) construct a new twisted linear period Λs,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∈Vπ (which we construct explicitly) for Λs,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]).