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 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, R) with a power savings error term. The proof requires novel techniques in the computation of the geometric side of the Kuznetsov trace formula.