Test Vectors for Archimedean Period Integrals

TL;DR

This paper demonstrates that newforms serve as weak test vectors for various archimedean period integrals related to $ ext{GL}_n$, extending the theory of test vectors from nonarchimedean to archimedean fields and applying these results to global automorphic periods.

Contribution

It establishes that newforms are weak test vectors for several archimedean period integrals, bridging archimedean and nonarchimedean theories and advancing the understanding of automorphic period integrals.

Findings

Newforms are weak test vectors for $ ext{GL}_n imes ext{GL}_n$ Rankin-Selberg integrals.

Newforms serve as weak test vectors for Flicker and Friedberg integrals.

Results enable the existence proof of weak test vectors for global automorphic period integrals.

Abstract

We study period integrals involving Whittaker functions associated to generic irreducible Casselman-Wallach representations of $\mathrm{GL}_n(F)$, where $F$ is an archimedean local field. Via the archimedean theory of newforms for $\mathrm{GL}_n$ developed by the first author, we prove that newforms are weak test vectors for several period integrals, including the $\mathrm{GL}_n \times \mathrm{GL}_n$ Rankin-Selberg integral, the Flicker integral, and the Bump-Friedberg integral. By taking special values of these period integrals, we deduce that newforms are weak test vectors for Rankin-Selberg periods, Flicker-Rallis periods, and Friedberg-Jacquet periods. These results parallel analogous results in the nonarchimedean setting proven by the second author, which use the nonarchimedean theory of newforms for $\mathrm{GL}_n$ developed by Jacquet, Piatetski-Shapiro, and Shalika. By combining these archimedean and nonarchimedean results, we prove the existence of weak test vectors for certain global period integrals of automorphic forms.