arXiv:1612.01109v1 [math.RT] 4 Dec 2016 The Ismagilov conjecture over a finite field F p A.V. Kosyak Max-Planck-Institut f¨ ur Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany Institute of Mathematics, Ukrainian National Academy of Sciences, 3 Tereshchenkivs’ka Str., Kyiv, 01601, Ukraine Abstract We construct the so-called quasiregular representations of the group of infi- nite upper triangular matrices with coefficients in a finite field and give the criteria of theirs irreducibility in terms of the initial measure. These repre- sentations are particular case of the Koopman representation hence, we find new conditions of its irreducibility. Since the field 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 Keywords: infinite-dimensional groups, finite field, unitary representation, irreducible representation, quasiregular representation, Koopman’s representation, Ismagilov’s conjecture, quasi-invariant, ergodic measure 2008 MSC: 22E65, (28C20, 43A80, 58D20) Email address: kosyak02@gmail.com (A.V. Kosyak ) Preprint submitted to March 14, 2019