An asymptotic orthogonality relation for GL(n, ℝ)

Goldfeld, Dorian; Stade, Eric; Woodbury, Michael

doi:10.2140/ant.2025.19.2185

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An asymptotic orthogonality relation for $\mathrm{GL}(n, \mathbb R)$

Dorian Goldfeld, Eric Stade and Michael Woodbury

Vol. 19 (2025), No. 11, 2185–2260

DOI: 10.2140/ant.2025.19.2185

Abstract

Orthogonality is a fundamental theme in representation theory and Fourier analysis. An orthogonality relation for characters of finite abelian groups (now recognized as an orthogonality relation on $GL (1)$ ) was used by Dirichlet to prove infinitely many primes in arithmetic progressions. Asymptotic orthogonality relations for $GL (n)$ , with $n \leq 3$ , and applications to number theory, have been considered by various researchers over the last 45 years. Recently, the authors of the present work have derived an explicit asymptotic orthogonality relation, with a power savings error term, for $GL (4, ℝ)$ . Here we extend those results to $GL (n, ℝ)$ , $n \geq 2$ .

For $n \leq 5$ , our results are contingent on the Ramanujan conjecture at the infinite place, but otherwise are unconditional. In particular, the case $n = 5$ represents a new result. The key new ingredient for the proof of the case $n = 5$ is the theorem of Kim and Shahidi that functorial products of cusp forms on $GL (2) \times GL (3)$ are automorphic on $GL (6)$ . For $n > 5$ (assuming again the Ramanujan conjecture holds at the infinite place), our results are conditional on two conjectures, both of which have been verified in various special cases. The first of these conjectures regards lower bounds for Rankin–Selberg L-functions, and the second concerns recurrence relations for Mellin transforms of $GL (n, ℝ)$ Whittaker functions.

Central to our proof is an application of the Kuznetsov trace formula, and a detailed analysis, utilizing a number of novel techniques, of the various entities — Hecke–Maass cusp forms, Langlands Eisenstein series, spherical principal series Whittaker functions and their Mellin transforms, and so on — that arise in this application.

Keywords

orthogonality, Hecke–Maass cusp forms, Kuznetsov trace formula

Mathematical Subject Classification

Primary: 11F55, 11F72

Milestones

Received: 16 November 2023

Revised: 9 September 2024

Accepted: 18 October 2024

Published: 14 September 2025

Authors

Dorian Goldfeld
Department of Mathematics Columbia University New York, NY United States

Eric Stade
Department of Mathematics University of Colorado Boulder Boulder, CO United States

Michael Woodbury
Department of Mathematics Rutgers University Piscataway, NJ United States

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Vol. 19, No. 11, 2025