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arXiv · 2112.06860

Test Vectors for Archimedean Period Integrals

Peter Humphries·Yeongseong Jo
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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.

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Peter Humphries, Yeongseong Jo. 2021-12-13. Test Vectors for Archimedean Period Integrals. https://doi.org/10.5565/publmat6812407

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