  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.
