Archimedean Newform Theory for ${\mathrm{GL}}_n$

Abstract

We introduce a new invariant, the conductor exponent, of a generic irreducible Casselman-Wallach representation of ${\mathrm{GL}}_n$ 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 ${\mathrm{GL}}_n\times {\mathrm{GL}}_n$ and ${\mathrm{GL}}_n\times {\mathrm{GL}}_{n-1}$ 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 ${\mathrm{GL}}_n$ over number fields. By-products of the proofs include new proofs of Stade's formulae and a new resolution of the test vector problem for archimedean Godement-Jacquet zeta integrals.

58 pages