An orthogonality relation for GL(4,R)

TL;DR

This paper establishes explicit orthogonality relations for the real group GL(4,R), advancing the understanding of harmonic analysis on higher rank groups with applications to number theory.

Contribution

It provides the first explicit orthogonality relations for GL(4,R) with a power savings error term, using novel techniques in trace formula analysis.

Findings

Derived explicit orthogonality relations for GL(4,R).

Achieved a power savings error term in the relations.

Provided new bounds for Kloosterman sums.

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. An appendix by Bingrong Huang gives new bounds for the relevant Kloosterman sums.