Geodesic
An orthogonality relation for $\mathrm {GL}(4, \mathbb R) $ (with an appendix by Bingrong Huang)
Forum of Mathematics, Sigma, Tome 9 (2021)

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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. 
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     title = {An orthogonality relation for $\mathrm {GL}(4, \mathbb R) $ (with an appendix by {Bingrong} {Huang)}},
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Dorian Goldfeld; Eric Stade; Michael Woodbury. An orthogonality relation for $\mathrm {GL}(4, \mathbb R) $ (with an appendix by Bingrong Huang). Forum of Mathematics, Sigma, Tome 9 (2021). doi: 10.1017/fms.2021.39

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