# Archimedean Non-vanishing, Cohomological Test Vectors, and Standard $L$-functions of $\mathrm{GL}_{2n}$: Complex Case

## TL;DR

This paper analyzes local zeta integrals for $ ext{GL}_{2n}( ext{C})$, establishing conditions for non-vanishing and constructing explicit test vectors, advancing the understanding of complex local $L$-functions and period integrals.

## Contribution

It provides necessary and sufficient conditions for non-zero cohomological test vectors and constructs explicit vectors, extending results from real to complex places.

## Key findings

- Characterizes when local Friedberg-Jacquet integrals are non-zero.
- Constructs explicit cohomological test vectors for complex places.
- Establishes conditions on characters for the existence of test vectors.

## 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 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 $v\in V_\pi$ (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.