The Ismagilov conjecture over a finite field ${\mathbb F}_p$

TL;DR

This paper constructs quasiregular representations of an infinite upper triangular matrix group over a finite field, analyzes their irreducibility, and revises the Ismagilov conjecture for finite fields.

Contribution

It introduces new irreducibility criteria for quasiregular representations over finite fields and corrects the Ismagilov conjecture in this context.

Findings

New irreducibility conditions for quasiregular representations

Identification of additional operators in the commutant due to finite field compactness

Correction of the Ismagilov conjecture for finite fields

Abstract

We construct the so-called quasiregular representations of the group $B_0^{\mathbb N}({\mathbb F}_p)$ of infinite upper triangular matrices with coefficients in a finite field and give the criteria of theirs irreducibility in terms of the initial measure. These representations are particular case of the Koopman representation hence, we find new conditions of its irreducibility. Since the field ${\mathbb F}_p$ is compact some new operators in the commutant emerges. Therefore, the Ismagilov conjecture in the case of the finite field should be corrected.