An orthogonality relation for $GL(4, \mathbb R) $ (with an appendix by Bingrong Huang) - Dorian Goldfeld; Eric Stade; Michael Woodbury (Forum of Mathematics, Sigma, 2021)
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. Orthogonality relations for $GL(2)$ and $GL(3)$ 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 $GL(4, \mathbb R)$ with a power savings error term. The proof requires novel techniques in the computation of the geometric side of the Kuznetsov trace formula.