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

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....

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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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ISSN2050-5094
2050-5094
DOI10.1017/fms.2021.39

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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.
AbstractList 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.
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.
ArticleNumber e47
Author Goldfeld, Dorian
Stade, Eric
Woodbury, Michael
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Cites_doi 10.1007/BF02570491
10.1006/jnth.1998.2301
10.4064/aa-80-1-1-48
10.1016/j.jfa.2020.108808
10.1090/S0894-0347-97-00220-8
10.1007/BF01406220
10.1155/S1073792895000225
10.24033/bsmf.1654
10.1090/pspum/066.2/1703764
10.1007/s00222-015-0583-y
10.4064/aa-78-4-405-409
10.1215/S0012-9074-02-11213-7
10.1093/imrn/rny061
10.4064/aa155-1-2
10.1007/s00222-013-0454-3
10.1073/pnas.34.5.204
10.1353/ajm.2001.0004
10.1007/s11139-013-9535-6
10.1007/s00209-018-2136-8
10.1017/CBO9780511470905
10.1142/S1793042115501031
10.1090/coll/058
10.1016/j.crma.2004.04.024
10.2307/2374461
10.1007/BF01390063
10.1007/BF01457276
10.4171/CMH/337
10.24033/bsmf.2201
10.2307/2118543
10.4153/CJM-1996-066-3
10.1007/BF01163653
10.1007/BFb0079929
10.1093/imrn/rnx087
10.1007/BF02784531
10.4064/aa-50-1-31-89
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References S2050509421000396_r38
S2050509421000396_r39
S2050509421000396_r33
S2050509421000396_r34
S2050509421000396_r35
S2050509421000396_r36
S2050509421000396_r30
S2050509421000396_r31
S2050509421000396_r32
S2050509421000396_r5
S2050509421000396_r4
S2050509421000396_r7
S2050509421000396_r6
S2050509421000396_r9
S2050509421000396_r48
S2050509421000396_r49
S2050509421000396_r44
S2050509421000396_r45
S2050509421000396_r46
S2050509421000396_r40
S2050509421000396_r41
S2050509421000396_r42
S2050509421000396_r43
S2050509421000396_r19
S2050509421000396_r15
S2050509421000396_r16
S2050509421000396_r17
Sarnak (S2050509421000396_r37) 1984; 70
S2050509421000396_r18
S2050509421000396_r11
S2050509421000396_r12
S2050509421000396_r13
S2050509421000396_r14
S2050509421000396_r51
S2050509421000396_r10
Deligne (S2050509421000396_r8) 1977
S2050509421000396_r50
S2050509421000396_r1
S2050509421000396_r26
S2050509421000396_r27
S2050509421000396_r3
S2050509421000396_r28
S2050509421000396_r29
S2050509421000396_r2
S2050509421000396_r22
S2050509421000396_r23
S2050509421000396_r24
S2050509421000396_r25
S2050509421000396_r20
S2050509421000396_r21
Vinogradov (S2050509421000396_r47) 1958; 22
References_xml – ident: S2050509421000396_r43
  doi: 10.1007/BF02570491
– volume: 70
  start-page: 321
  volume-title: Statistical properties of eigenvalues of the Hecke operators
  year: 1984
  ident: S2050509421000396_r37
– ident: S2050509421000396_r10
  doi: 10.1006/jnth.1998.2301
– ident: S2050509421000396_r9
  doi: 10.4064/aa-80-1-1-48
– ident: S2050509421000396_r42
  doi: 10.1016/j.jfa.2020.108808
– ident: S2050509421000396_r20
– ident: S2050509421000396_r39
  doi: 10.1090/S0894-0347-97-00220-8
– ident: S2050509421000396_r5
  doi: 10.1007/BF01406220
– ident: S2050509421000396_r26
  doi: 10.1155/S1073792895000225
– ident: S2050509421000396_r27
  doi: 10.24033/bsmf.1654
– volume: 22
  start-page: 161
  year: 1958
  ident: S2050509421000396_r47
  article-title: A new estimate of the function $\zeta \left(1+\mathrm{it}\right)$
  publication-title: Izv. Akad. Nauk SSSR. Ser. Mat.
– ident: S2050509421000396_r32
  doi: 10.1090/pspum/066.2/1703764
– ident: S2050509421000396_r41
  doi: 10.1007/s00222-015-0583-y
– ident: S2050509421000396_r7
  doi: 10.4064/aa-78-4-405-409
– ident: S2050509421000396_r16
– ident: S2050509421000396_r33
  doi: 10.1215/S0012-9074-02-11213-7
– ident: S2050509421000396_r6
  doi: 10.1093/imrn/rny061
– ident: S2050509421000396_r15
  doi: 10.4064/aa155-1-2
– ident: S2050509421000396_r4
  doi: 10.1007/s00222-013-0454-3
– ident: S2050509421000396_r49
  doi: 10.1073/pnas.34.5.204
– ident: S2050509421000396_r44
  doi: 10.1353/ajm.2001.0004
– ident: S2050509421000396_r38
– ident: S2050509421000396_r51
  doi: 10.1007/s11139-013-9535-6
– ident: S2050509421000396_r24
  doi: 10.1007/s00209-018-2136-8
– ident: S2050509421000396_r19
– ident: S2050509421000396_r31
– ident: S2050509421000396_r50
– ident: S2050509421000396_r12
– ident: S2050509421000396_r36
  doi: 10.1017/CBO9780511470905
– ident: S2050509421000396_r23
  doi: 10.1142/S1793042115501031
– ident: S2050509421000396_r40
  doi: 10.1090/coll/058
– ident: S2050509421000396_r18
  doi: 10.1016/j.crma.2004.04.024
– ident: S2050509421000396_r28
– ident: S2050509421000396_r34
  doi: 10.2307/2374461
– ident: S2050509421000396_r22
– ident: S2050509421000396_r48
  doi: 10.1007/BF01390063
– ident: S2050509421000396_r46
  doi: 10.1007/BF01457276
– ident: S2050509421000396_r2
  doi: 10.4171/CMH/337
– ident: S2050509421000396_r14
– ident: S2050509421000396_r11
  doi: 10.24033/bsmf.2201
– start-page: 154
  volume-title: Dualité
  year: 1977
  ident: S2050509421000396_r8
– ident: S2050509421000396_r1
– ident: S2050509421000396_r25
  doi: 10.2307/2118543
– ident: S2050509421000396_r35
– ident: S2050509421000396_r29
  doi: 10.4153/CJM-1996-066-3
– ident: S2050509421000396_r13
  doi: 10.1007/BF01163653
– ident: S2050509421000396_r30
  doi: 10.1007/BFb0079929
– ident: S2050509421000396_r21
– ident: S2050509421000396_r17
  doi: 10.1093/imrn/rnx087
– ident: S2050509421000396_r45
  doi: 10.1007/BF02784531
– ident: S2050509421000396_r3
  doi: 10.4064/aa-50-1-31-89
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Snippet Orthogonality is a fundamental theme in representation theory and Fourier analysis. An orthogonality relation for characters of finite abelian groups (now...
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SubjectTerms Dirichlet problem
Fourier analysis
Group theory
Number theory
Numbers
Orthogonality
Progressions
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