TEST VECTORS FOR ARCHIMEDEAN PERIOD INTEGRALS

We study period integrals involving Whittaker functions associated to generic irreducible Casselman-Wallach representations of GL_n(F), where F is an archimedean local held. Via the archimedean theory of newforms for GL_n developed by the first author, we prove that newforms are weak test vectors for several period integrals, including the GL_n × 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 proved by the second author, which use the nonarchimedean theory of newforms for 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.