Archimedean Asai Zeta Integrals and Cohomological Test Vectors
Abstract
We construct explicit test vectors in the minimal U(n)-types of cohomological principal series representations of GLn(ℂ), together with Schwartz functions, for which the Flicker zeta integrals equal the corresponding Asai L-factors. Both the vectors and the Schwartz functions are independent of s, with all normalization constants given explicitly. We construct the vectors through a polynomial realization of Schur modules. Schur orthogonality reduces the Flicker integrals to Mellin transforms of Whittaker functions, and Godement sections yield a recurrence for their Jacquet integrals. The lower-rank vectors depend on the remaining integration variables, preventing a direct reduction to lower-rank Flicker integrals. We overcome this difficulty by combining growth estimates derived from the recurrence with the Asai functional equation. This proves that the holomorphic quotient of the Flicker integral by its Asai L-factor is constant. An explicit asymptotic calculation shows that this constant is exactly one.
Related
- Archimedean zeta integrals on $ {\rm SO}_{2n} \times {\rm GL}_n$ and $ {\rm Sp}_{2n} \times {\rm GL}_n $
- Exceptional poles of archimedean Asai L-functions
- Exact Archimedean Flicker--Rallis Periods on Cohomological Minimal K-Types
- Shifted log-sine integrals and multiple zeta values
- A Unified Approach to Formal Arakawa-Kaneko Type Zeta Values via Iterated Integrals
BibTeX
@misc{lin2026archimedean,
title = {{Archimedean Asai Zeta Integrals and Cohomological Test Vectors}},
author = {Bingchen Lin and Dongming She},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38191},
eprint = {2610.38191},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38191},
}
RIS
TY - UNPB
TI - Archimedean Asai Zeta Integrals and Cohomological Test Vectors
AU - Bingchen Lin
AU - Dongming She
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38191
UR - https://arxiv.org/abs/2610.38191
AB - We construct explicit test vectors in the minimal U(n)-types of cohomological principal series representations of GL_n(ℂ), together with Schwartz functions, for which the Flicker zeta integrals equal the corresponding Asai L-factors. Both the vectors and the Schwartz functions are independent of s, with all normalization constants given explicitly. We construct the vectors through a polynomial realization of Schur modules. Schur orthogonality reduces the Flicker integrals to Mellin transforms of Whittaker functions, and Godement sections yield a recurrence for their Jacquet integrals. The lower-rank vectors depend on the remaining integration variables, preventing a direct reduction to lower-rank Flicker integrals. We overcome this difficulty by combining growth estimates derived from the recurrence with the Asai functional equation. This proves that the holomorphic quotient of the Flicker integral by its Asai L-factor is constant. An explicit asymptotic calculation shows that this constant is exactly one.
ER -