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The fine spectral expansion of the Rankin-Selberg period

Paul Boisseau

TL;DR

This work establishes the fine spectral expansion for the Rankin--Selberg period attached to the Rankin--Selberg spherical variety $X=G/H$ with $G= ext{GL}_n imes ext{GL}_{n+1}$ and $H= ext{GL}_n$. The authors develop a robust framework of regularized Rankin--Selberg periods, extending beyond Arthur-type representations to include non-tempered and residual contributions, and prove a spectral expansion via careful contour-shifting arguments à la Langlands, together with precise control of Eisenstein series and their residues. A key innovation is the regularization of periods for general inducing data $(I,P, ho)$, the parabolic-descent description of these periods, and the analysis of their poles, residues, and functional equations, culminating in an explicit expansion of $J^H(g,f)$ as an integral over unitary contours of relative characters. The results provide a crucial ingredient toward the Jacquet--Rallis trace formula’s fine spectral expansion, with implications for non-tempered Gan–Gross–Prasad-type conjectures and related L-value phenomena. The paper also furnishes detailed bounds for discrete Eisenstein series on $ ext{GL}_n$ and develops the machinery to extend Langlands’ spectral decomposition to broader spaces of automorphic functions, enabling precise Euler-product expressions for regularized periods.

Abstract

We state and prove the spectral expansion of the theta series attached to the Rankin-Selberg spherical variety $(\mathrm{GL}_{n+1} \times \mathrm{GL}_n)/\mathrm{GL}_n$. This is a key result towards the fine spectral expansion of the Jacquet-Rallis trace formula. Our expansion is written in terms of regularized Rankin--Selberg periods for non-tempered automorphic representations, which we show compute special values of $L$-functions. The proof relies on shifts of contours of integration à la Langlands. We also establish two technical but crucial results on bounds and singularities for discrete Eisenstein series of $\mathrm{GL}_n$ in the positive Weyl chamber.

The fine spectral expansion of the Rankin-Selberg period

TL;DR

This work establishes the fine spectral expansion for the Rankin--Selberg period attached to the Rankin--Selberg spherical variety with and . The authors develop a robust framework of regularized Rankin--Selberg periods, extending beyond Arthur-type representations to include non-tempered and residual contributions, and prove a spectral expansion via careful contour-shifting arguments à la Langlands, together with precise control of Eisenstein series and their residues. A key innovation is the regularization of periods for general inducing data , the parabolic-descent description of these periods, and the analysis of their poles, residues, and functional equations, culminating in an explicit expansion of as an integral over unitary contours of relative characters. The results provide a crucial ingredient toward the Jacquet--Rallis trace formula’s fine spectral expansion, with implications for non-tempered Gan–Gross–Prasad-type conjectures and related L-value phenomena. The paper also furnishes detailed bounds for discrete Eisenstein series on and develops the machinery to extend Langlands’ spectral decomposition to broader spaces of automorphic functions, enabling precise Euler-product expressions for regularized periods.

Abstract

We state and prove the spectral expansion of the theta series attached to the Rankin-Selberg spherical variety . This is a key result towards the fine spectral expansion of the Jacquet-Rallis trace formula. Our expansion is written in terms of regularized Rankin--Selberg periods for non-tempered automorphic representations, which we show compute special values of -functions. The proof relies on shifts of contours of integration à la Langlands. We also establish two technical but crucial results on bounds and singularities for discrete Eisenstein series of in the positive Weyl chamber.
Paper Structure (139 sections, 86 theorems, 439 equations)

This paper contains 139 sections, 86 theorems, 439 equations.

Table of Contents

  1. Introduction
  2. Motivations
  3. Spectral decomposition of the Rankin--Selberg spherical variety
  4. The Jacquet--Rallis relative trace formula
  5. The fine spectral expansion of the Rankin--Selberg period
  6. Preliminary notations
  7. Relevant inducing data
  8. Regularized periods
  9. The fine spectral expansion
  10. About Theorem \ref{['thm:spectral_expansion_intro']}
  11. Step 1: coarse unfolding of the Rankin--Selberg integral
  12. Step 2: residues of Eisenstein series
  13. Step 3: residues of regularized periods
  14. Outline of the paper
  15. Acknowledgement
  16. ...and 124 more sections

Key Result

Theorem 1.4

Let $(I,P,\pi) \in \Pi_H$. The following assertions hold.

Theorems & Definitions (172)

  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Theorem 1.4
  • Example 1.5
  • Theorem 1.6
  • Remark 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 162 more