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.
Some Results in the Fourier Analysis
