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 on $\mathrm {GL}(1)$ ) was used by Dirichlet to prove infinitely many primes arithmetic progressions. relations {GL}(2)$ {GL}(3)$ have been worked researchers with broad range applications number theory. We present here, the first time, very explicit real group {GL}(4, \mathbb R)$ power savings error term. The proof requires novel techniques computation geometric side Kuznetsov trace formula.