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

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.