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

Peter Humphries

TL;DR

This work develops an archimedean analogue of the classical nonarchimedean newform theory for GL_n, introducing the conductor exponent c(π) and the newform K-type τ∘ as the minimal-degree K-type with a unique K_{n-1}-invariant line. It proves that the newform is unique (multiplicity one), and that c(π) is additive under isobaric sums and inductive under Langlands-type constructions; it further establishes that the archimedean newform serves as a test vector for GL_n×GL_m Rankin–Selberg integrals (with the second factor spherical in many cases) and for Godement–Jacquet zeta integrals. The paper provides three constructive descriptions of the newform—via the Iwasawa decomposition, convolution sections, and Godement sections—along with explicit models using homogeneous harmonic polynomials, enabling closed formulas and propagation relations for the associated Whittaker functions. These results unify archimedean and nonarchimedean perspectives, yield streamlined proofs of Stade-type integrals, and lay groundwork for global Eulerian integrals in automorphic contexts.

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.

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

TL;DR

This work develops an archimedean analogue of the classical nonarchimedean newform theory for GL_n, introducing the conductor exponent c(π) and the newform K-type τ∘ as the minimal-degree K-type with a unique K_{n-1}-invariant line. It proves that the newform is unique (multiplicity one), and that c(π) is additive under isobaric sums and inductive under Langlands-type constructions; it further establishes that the archimedean newform serves as a test vector for GL_n×GL_m Rankin–Selberg integrals (with the second factor spherical in many cases) and for Godement–Jacquet zeta integrals. The paper provides three constructive descriptions of the newform—via the Iwasawa decomposition, convolution sections, and Godement sections—along with explicit models using homogeneous harmonic polynomials, enabling closed formulas and propagation relations for the associated Whittaker functions. These results unify archimedean and nonarchimedean perspectives, yield streamlined proofs of Stade-type integrals, and lay groundwork for global Eulerian integrals in automorphic contexts.

Abstract

We introduce a new invariant, the conductor exponent, of a generic irreducible Casselman-Wallach representation of 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 and 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 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.

Paper Structure

This paper contains 75 sections, 66 theorems, 279 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Groups and Haar Measures
  4. Local Fields and Absolute Values
  5. Haar Measures on $F$ and $F^{\times}$
  6. Subgroups of $\mathop{\mathrm{GL}}\nolimits_n(F)$ and the Iwasawa Decomposition
  7. Haar Measures on $\mathop{\mathrm{GL}}\nolimits_n(F)$ and Its Subgroups
  8. Representations
  9. Isobaric Sums
  10. Induced Representations of Whittaker and Langlands Types
  11. Whittaker Models
  12. Irreducible Representations
  13. Spherical Representations
  14. Integral Representations of L-Functions
  15. Rankin--Selberg Integrals
  16. ...and 60 more sections

Key Result

Theorem 3.1

Let $(\pi,V_{\pi})$ be an induced representation of Langlands type of $\mathop{\mathrm{GL}}\nolimits_n(F)$. There exists a minimal nonnegative integer $m$ for which $V_{\pi}^{K_1(\mathfrak{p}^m)}$ is nontrivial. For this minimal value of $m$, $V_{\pi}^{K_1(\mathfrak{p}^m)}$ is one-dimensional.

Theorems & Definitions (140)

  • Theorem 3.1: Jacquet--Piatetski-Shapiro--Shalika JP-SS81
  • Definition 3.2
  • Remark 3.3
  • Remark 3.4
  • Theorem 3.5: Hum22
  • Definition 3.6
  • Theorem 3.7: Reeder Ree91
  • Theorem 3.8: Hum22
  • Theorem 3.9: Jacquet--Piateski-Shapiro--Shalika JP-SS83
  • Theorem 3.10: Jacquet--Piateski-Shapiro--Shalika JP-SS83
  • ...and 130 more