On equidistribution of Gauss sums of cuspidal representations of $GL_d(\mathbb F_q)$

TL;DR

This paper studies the distribution of Gauss sum angles for cuspidal representations of finite general linear groups, showing equidistribution in general but clustering in specific cases like PGL_2.

Contribution

It demonstrates the equidistribution of Gauss sum angles for cuspidal representations of GL_d over finite fields and describes their clustering behavior in PGL_2 cases.

Findings

Gauss sum angles are equidistributed for cuspidal representations of GL_d.

Angles cluster around 1 and -1 for PGL_2 over finite fields with odd p.

Angles cluster around 1 for PGL_2 over fields with p=2.

Abstract

We investigate the distribution of the angles of Gauss sums attached to the cuspidal representations of general linear groups over finite fields. In particular we show that they happen to be equidistributed w.r.t.the Haar measure. However, for representations of $PGL_2(\mathbb F_q)$, they are clustered around $1$ and $-1$ for odd $p$ and around $1$ for $p=2$.