An orthogonality relation for (with an appendix by Bingrong Huang)

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 $\mathrm {GL}(1)$ ) was used by Dirichlet to prove infinitely many primes in arithmetic progr...

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Bibliographic Details
Published inForum of mathematics. Sigma Vol. 9
Main Authors Goldfeld, Dorian, Stade, Eric, Woodbury, Michael
Format Journal Article
LanguageEnglish
Published Cambridge Cambridge University Press 2021
Subjects
Dirichlet problem
Fourier analysis
Group theory
Number theory
Numbers
Orthogonality
Progressions
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Summary: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 $\mathrm {GL}(1)$ ) was used by Dirichlet to prove infinitely many primes in arithmetic progressions. Orthogonality relations for $\mathrm {GL}(2)$ and $\mathrm {GL}(3)$ have been worked on by many researchers with a broad range of applications to number theory. We present here, for the first time, very explicit orthogonality relations for the real group $\mathrm {GL}(4, \mathbb R)$ with a power savings error term. The proof requires novel techniques in the computation of the geometric side of the Kuznetsov trace formula.
ISSN:2050-5094
2050-5094
DOI:10.1017/fms.2021.39