Regular characters of $GL_n(O)$ and Weil representations over finite fields

TL;DR

This paper addresses a gap in a previous proof related to regular characters of $GL_n(O)$ and Weil representations over finite fields, introducing new arguments using Schr"odinger and Weil representations to extend results.

Contribution

It provides a new proof approach for regular characters of $GL_n(O)$ over finite fields, covering cuspidal and split cases, and discusses a conjecture on Schur multipliers.

Findings

Identifies a gap in G. Hill's theorem proof.

Introduces Schr"odinger and Weil representations for new arguments.

Extends results to regular characters including cuspidal cases.

Abstract

In this paper, we will point out a gap in the proof of a theorem of G.Hill (J. Algebra, 174 (1995), 610-635) and will give new arguments to give a remedy in the non-dyadic case modulo a conjecture on the triviality of certain Schur multiplier associated with a symplectic space over finite field. The new argument uses the Schr\"odinger representation of the Heisenberg group associated with a symplectic space over a finite field, and a simple application of Weil representation. This argument is applicable to the regular characters in general which include the cuspidal cases as well as the regular split cases.