Weighted Sato-Tate Vertical Distribution of the Satake Parameter of Maass Forms on PGL(N)

TL;DR

This paper formulates a conjecture on the orthogonality of Fourier coefficients of Maass forms on PGL(N), and proves a weighted vertical equidistribution theorem for Satake parameters at finite primes, extending previous work for N=3.

Contribution

It introduces a new conjecture on Fourier coefficient orthogonality for N≥4 and proves a weighted equidistribution theorem for Satake parameters, generalizing prior results for N=3.

Findings

Proved weighted vertical equidistribution theorem for Satake parameters.

Established a conjectured orthogonality relation for Fourier coefficients.

Derived convergence rates for N=3 case.

Abstract

We formulate a conjectured orthogonality relation between the Fourier coefficients of Maass forms on PGL(N) for N>=2. Based on the work of Goldfeld-Kontorovich and Blomer for N=3, and on our conjecture for N>=4, we prove a weighted vertical equidistribution theorem (with respect to the generalized Sato-Tate measure) for the Satake parameter of Maass forms at a finite prime. For N=3, the rate of convergence for the equidistribution theorem is obtained.