Archimedean Newform Theory For

Abstract We introduce a new invariant, the conductor exponent, of a generic irreducible Casselman–Wallach representation of {GL}ₙ (F), where F is an archimedean local field, that quantifies the extent to which this representation may be ramified. We also determine a distinguished vector, the newform, occurring with multiplicity one in this representation, with the complexity of this vector measured in a natural way by the conductor exponent. Finally, we show that the newform is a test vector for {GL}ₙ {GL}ₙ and {GL}ₙ {GL}₍ - ₁ Rankin–Selberg integrals when the second representation is unramified. This theory parallels an analogous nonarchimedean theory due to Jacquet, Piatetski-Shapiro, and Shalika; combined, this completes a global theory of newforms for automorphic representations of {GL}ₙ over number fields. By-products of the proofs include new proofs of Stade’s formulæ and a new resolution of the test vector problem for archimedean Godement–Jacquet zeta integrals.