An Asymptotic Orthogonality Relation for ${\rm GL}(n, \mathbb R)$

Dorian Goldfeld, Eric Stade, Michael Woodbury

Published 2022-12-30Version 1

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. Asymptotic orthogonality relations for GL(n), with $n\le 4$, and applications to number theory, have been considered by various researchers. Here we present an explicit, asymptotic orthogonality relation for GL(n, R) ($n\ge2$), with a power savings error term. The proof requires novel techniques in the computation of the geometric side of the Kuznetsov trace formula.